🧠:[<think>Okay, so I have this problem here where a is given as 1 divided by (2 plus the square root of 3). And I need to find the value of the square root of (a squared minus 2a plus 1). The options are given, so maybe I can check them after simplifying, but let me try to solve it step by step.First, let's write down what's given. a = 1/(2 + √3). Hmm, that denominator has a radical, so maybe I should rationalize it first? Yeah, rationalizing the denominator might make things easier later on. To rationalize 1/(2 + √3), I can multiply the numerator and denominator by the conjugate of the denominator, which is (2 - √3). Let's do that.So, multiplying numerator and denominator by (2 - √3):a = [1 * (2 - √3)] / [(2 + √3)(2 - √3)]Let me compute the denominator first. Using the difference of squares formula: (a + b)(a - b) = a² - b². Here, a is 2 and b is √3. So:(2 + √3)(2 - √3) = 2² - (√3)² = 4 - 3 = 1.Oh, that's nice! So the denominator becomes 1, which simplifies things. Therefore, a = (2 - √3)/1 = 2 - √3. So a is 2 minus √3. Okay, that's simpler.Now, the expression we need is sqrt(a² - 2a + 1). Let's look at that expression inside the square root: a² - 2a + 1. Hmm, that looks familiar. Wait, that's the expansion of (a - 1)². Because (a - 1)² = a² - 2a + 1. Yes, exactly. So sqrt(a² - 2a +1) is sqrt((a - 1)²), which is |a - 1|. Since the square root of a square is the absolute value.So now, the problem reduces to finding |a - 1|. Let's compute a - 1 first. We know a is 2 - √3, so:a - 1 = (2 - √3) - 1 = 1 - √3.Then the absolute value of that is |1 - √3|. Now, √3 is approximately 1.732, so 1 - 1.732 is about -0.732, which is negative. Therefore, the absolute value would make it positive, so |1 - √3| = √3 - 1.Looking at the options, (A) is √3 - 1 and (B) is 1 - √3. Since we have an absolute value, the answer should be the positive one, which is (A) √3 - 1.Wait, but let me double-check to make sure I didn't make a mistake in simplifying. Let me go through the steps again.First, a = 1/(2 + √3). Multiply numerator and denominator by (2 - √3) to get a = (2 - √3)/[(2 + √3)(2 - √3)] = (2 - √3)/1 = 2 - √3. That's correct.Then, the expression sqrt(a² - 2a + 1). Recognize that as sqrt((a - 1)^2) = |a - 1|. Then a - 1 = (2 - √3) - 1 = 1 - √3. The absolute value of that is √3 - 1 because 1 - √3 is negative. So yes, the answer should be (A).Alternatively, maybe I can compute the expression without simplifying first. Let's see.Compute a² - 2a + 1. Let's compute a first as 1/(2 + √3). Let's compute a²:a² = [1/(2 + √3)]² = 1/(2 + √3)^2.Then 2a = 2/(2 + √3).So the expression is 1/(2 + √3)^2 - 2/(2 + √3) + 1.Hmm, that seems more complicated. Let's see if we can compute this.First, compute (2 + √3)^2: (2)^2 + 2*2*√3 + (√3)^2 = 4 + 4√3 + 3 = 7 + 4√3.So a² = 1/(7 + 4√3).Then 2a = 2/(2 + √3).So the expression becomes:1/(7 + 4√3) - 2/(2 + √3) + 1.This seems messy. Let me see if I can rationalize these denominators.First term: 1/(7 + 4√3). Multiply numerator and denominator by (7 - 4√3):1*(7 - 4√3)/[(7 + 4√3)(7 - 4√3)] = (7 - 4√3)/(49 - (4√3)^2).Compute denominator: 49 - 16*3 = 49 - 48 = 1. So first term is (7 - 4√3)/1 = 7 - 4√3.Second term: -2/(2 + √3). Multiply numerator and denominator by (2 - √3):-2*(2 - √3)/[(2 + √3)(2 - √3)] = -2*(2 - √3)/1 = -4 + 2√3.Third term is +1.Now add all three terms:First term: 7 - 4√3Second term: -4 + 2√3Third term: +1Adding together:(7 - 4 + 1) + (-4√3 + 2√3) = (4) + (-2√3) = 4 - 2√3.So the expression inside the sqrt is 4 - 2√3. Wait, but then sqrt(4 - 2√3). Is that equal to √3 -1?Wait, let me check. Let's compute (√3 -1)^2. That's (√3)^2 - 2*√3*1 +1^2 = 3 - 2√3 +1 = 4 - 2√3. Yes! So sqrt(4 -2√3) is sqrt((√3 -1)^2) = |√3 -1|. Since √3 is approximately 1.732, √3 -1 is positive, so absolute value is √3 -1. Therefore, the answer is (A). So that's another way to confirm it.Wait, so even if I compute the original expression without recognizing the perfect square, I end up with sqrt(4 -2√3) which is equal to √3 -1. So both methods lead to the same answer, which is (A). So that seems solid.Alternatively, maybe I can check the answer by substituting the options. Let's see. If the answer is (A) sqrt3 -1, then squaring that should give 4 - 2√3, as we saw earlier. And (B) is 1 - sqrt3, squaring that would also give 1 -2√3 +3 =4 -2√3, but since sqrt(a^2 -2a +1) is |a -1|, which is |1 - sqrt3| which is sqrt3 -1. So yeah, answer (A). So options (A) and (B) when squared both give 4 -2√3, but since we need the square root, we take the positive one.Alternatively, let me check the value numerically to confirm. Let me compute a first.Given a =1/(2 +√3). Let's approximate √3 ≈1.732. Then denominator is 2 +1.732≈3.732. So a≈1/3.732≈0.2679. Then compute sqrt(a² -2a +1). Let's compute a²: 0.2679²≈0.0718. Then -2a: -2*0.2679≈-0.5358. So 0.0718 -0.5358 +1≈0.0718 -0.5358= -0.464 +1=0.536. Then sqrt(0.536)≈0.732.Now, let's compute the options. (A) √3 -1≈1.732 -1=0.732. (B) 1 -√3≈-0.732. (C) 1/(1 +√3)≈1/2.732≈0.366. (D) is 1/(1 -√3)≈1/(-0.732)≈-1.366. So the value we computed numerically is approximately 0.732, which matches (A) √3 -1≈0.732. So that's another confirmation.Therefore, the answer should be (A).Final Answerboxed{A}</think>Given ( a = frac{1}{2 + sqrt{3}} ), we need to find the value of ( sqrt{a^2 - 2a + 1} ).First, we rationalize the denominator of ( a ):[a = frac{1}{2 + sqrt{3}} cdot frac{2 - sqrt{3}}{2 - sqrt{3}} = frac{2 - sqrt{3}}{(2 + sqrt{3})(2 - sqrt{3})} = frac{2 - sqrt{3}}{1} = 2 - sqrt{3}]Next, we recognize that the expression inside the square root is a perfect square:[sqrt{a^2 - 2a + 1} = sqrt{(a - 1)^2} = |a - 1|]We compute ( a - 1 ):[a - 1 = (2 - sqrt{3}) - 1 = 1 - sqrt{3}]Taking the absolute value, we get:[|1 - sqrt{3}| = sqrt{3} - 1]Thus, the value of ( sqrt{a^2 - 2a + 1} ) is (boxed{A}).,<think>Okay, let's see. The problem is asking for the value of √(a² - 2a + 1) where a is given as 1/(2 + √3). The options are A to D. Hmm. Alright, first step is to simplify the expression inside the square root. Let me write that down.So, the expression is √(a² - 2a + 1). Wait a minute, that looks familiar. Isn't a² - 2a + 1 a perfect square? Yeah, it's the expansion of (a - 1)². So, the square root of (a - 1) squared. But since square roots of squares can be tricky, we have to consider the absolute value. So, √(a² - 2a + 1) = |a - 1|. That simplifies things. So instead of dealing with the quadratic expression, we can just compute |a - 1|.Alright, so now I need to find the value of |a - 1| where a = 1/(2 + √3). Let's compute a first. Maybe rationalize the denominator? Because 1 over 2 + √3 is a bit messy. To rationalize, multiply numerator and denominator by the conjugate, which is 2 - √3. So:a = 1/(2 + √3) * (2 - √3)/(2 - √3) = (2 - √3)/[(2 + √3)(2 - √3)]Calculating the denominator: (2 + √3)(2 - √3) = 2² - (√3)² = 4 - 3 = 1. Oh, that's nice. So denominator is 1, so a = 2 - √3. Wait, that's super helpful. So a simplifies to 2 - √3.Wait, so a = 2 - √3. Then a - 1 is (2 - √3) - 1 = 1 - √3. Then |a - 1| is |1 - √3|. But √3 is approximately 1.732, so 1 - √3 is approximately -0.732. The absolute value of that is √3 - 1. So, |1 - √3| = √3 - 1. Therefore, the value of the original expression is √3 - 1, which is option A.But let me double-check in case I made a mistake. Let's verify step by step.First, a = 1/(2 + √3). Rationalizing:Multiply numerator and denominator by 2 - √3:(1*(2 - √3)) / [(2 + √3)(2 - √3)].Denominator: 2*2 + 2*(-√3) + √3*2 + √3*(-√3) = 4 - (√3)^2 = 4 - 3 = 1. So denominator is 1. So numerator is 2 - √3, so a = 2 - √3. Correct.Then, a - 1 = 2 - √3 - 1 = 1 - √3. Absolute value of that is |1 - √3|. Since √3 > 1, this is √3 - 1. So the square root of (a² - 2a + 1) is √3 - 1. Therefore, the answer is A.But wait, let me check if there's another way to approach this. Maybe instead of simplifying a first, plug a into the original expression. Let's see:Compute √(a² - 2a + 1). Since this is |a - 1|, as we established, so maybe that's the same as |1 - a|. But since absolute value is symmetric, |a - 1| = |1 - a|. Either way, same result.Alternatively, maybe compute a² - 2a + 1 directly. Let's try that.First, a = 1/(2 + √3). Compute a²:a² = [1/(2 + √3)]² = 1/(2 + √3)^2. Let's compute denominator: (2 + √3)^2 = 4 + 4√3 + 3 = 7 + 4√3. So a² = 1/(7 + 4√3).Then, -2a = -2*(1/(2 + √3)) = -2/(2 + √3).So putting together:a² - 2a + 1 = [1/(7 + 4√3)] - [2/(2 + √3)] + 1.This seems more complicated. Let's try to rationalize each term:First term: 1/(7 + 4√3). Multiply numerator and denominator by conjugate 7 - 4√3:[1*(7 - 4√3)] / [(7 + 4√3)(7 - 4√3)] = (7 - 4√3)/(49 - (4√3)^2).Compute denominator: 49 - 16*3 = 49 - 48 = 1. So first term is 7 - 4√3.Second term: -2/(2 + √3). Multiply numerator and denominator by 2 - √3:[-2*(2 - √3)] / [(2 + √3)(2 - √3)] = [-4 + 2√3]/1 = -4 + 2√3.Third term is 1.So adding them all together:First term: 7 - 4√3Second term: -4 + 2√3Third term: +1Total: (7 - 4 + 1) + (-4√3 + 2√3) = 4 - 2√3.Wait, so a² - 2a + 1 = 4 - 2√3? Then √(4 - 2√3). Hmm, but this contradicts our previous conclusion. Wait, so which one is correct?Wait, hold on, if a = 2 - √3, then a - 1 = 1 - √3, whose absolute value is √3 -1, so the expression is √3 -1. However, when I computed a² - 2a + 1 directly, I ended up with 4 - 2√3. Let me check if these two are equivalent.Is (√3 -1)^2 equal to 4 - 2√3?Compute (√3 -1)^2 = (√3)^2 - 2*√3*1 +1 = 3 - 2√3 +1 = 4 - 2√3. Yes! So (√3 -1)^2 = 4 - 2√3. Therefore, √(4 - 2√3) = √[(√3 -1)^2] = |√3 -1| = √3 -1, since √3 >1. Therefore, both methods confirm that the value is √3 -1. So answer is A.But why when I directly compute, I get 4 - 2√3 inside the square root? Because a² -2a +1 is (a -1)^2, which when a =2 -√3, gives (2 -√3 -1)^2 = (1 -√3)^2 = (√3 -1)^2, which is 4 -2√3. Then taking square root gives |√3 -1| = √3 -1. So both methods are consistent.Therefore, the answer is A. So I think that's correct.But let me check again the options:(A) √3 −1(B) 1−√3(C) 1/(1+√3)(D) 1/(1−√3)Since we have √(a² -2a +1) = |a -1| = |1 - a| = √3 -1. So A is the answer.Alternatively, if I miscalculated during the direct computation, but cross-verifying, both methods give the same result. So I think the answer is A.But let me check once more. Let's compute a again. a =1/(2 +√3). Rationalizing:Multiply numerator and denominator by 2 -√3:a = [1*(2 -√3)] / [(2 +√3)(2 -√3)] = (2 -√3)/(4 -3) = (2 -√3)/1 = 2 -√3. Correct.So a =2 -√3. Then a -1 =2 -√3 -1=1 -√3. Absolute value is |1 -√3|=√3 -1. So answer is A. Definitely. Therefore, confident that the answer is A.Final Answerboxed{A}</think>Given ( a = frac{1}{2 + sqrt{3}} ), we need to find the value of ( sqrt{a^2 - 2a + 1} ).First, we simplify ( a ). Rationalizing the denominator:[a = frac{1}{2 + sqrt{3}} cdot frac{2 - sqrt{3}}{2 - sqrt{3}} = frac{2 - sqrt{3}}{(2 + sqrt{3})(2 - sqrt{3})} = frac{2 - sqrt{3}}{4 - 3} = 2 - sqrt{3}]Next, we recognize that ( sqrt{a^2 - 2a + 1} ) is the square root of a perfect square:[sqrt{a^2 - 2a + 1} = sqrt{(a - 1)^2} = |a - 1|]Substituting ( a = 2 - sqrt{3} ):[a - 1 = (2 - sqrt{3}) - 1 = 1 - sqrt{3}]Taking the absolute value:[|1 - sqrt{3}| = sqrt{3} - 1]Thus, the value of ( sqrt{a^2 - 2a + 1} ) is ( sqrt{3} - 1 ), which corresponds to option A.[boxed{A}]]

🔑:2. A.Notice that a=frac{1}{2+sqrt{3}}=2-sqrt{3}<1.Then sqrt{a^{2}-2 a+1}=sqrt{(a-1)^{2}} =1-a=sqrt{3}-1.

🧠:[<think>Okay, let's see. The problem is about four distinct positive integers a, b, c, d such that their product abcd is 441. Then we have to find the sum a + b + c + d from the given options: 30, 32, 31, 36. The choices are A to D. Alright, let's start by breaking down what 441 factors into because the product of the four numbers is 441.First, I remember that 441 is 21 squared, right? 21 times 21 is 441. But 21 itself is 3 times 7. So 441 is 3^2 * 7^2. Let me verify that: 3 squared is 9, 7 squared is 49, and 9 times 49 is 441. Yep, that's correct. So the prime factorization of 441 is 3² * 7². Now, we need four distinct positive integers whose product is 441. Since the prime factors are 3, 3, 7, 7, we need to distribute these prime factors into four different numbers. But all four numbers have to be distinct. Hmm, that might be a bit tricky because we have repeated prime factors here.Let me think. If we have to split the factors 3² * 7² into four distinct integers, maybe we can start by considering the factors of 441 and pick four distinct ones that multiply to 441. Alternatively, since the prime factors are limited, maybe we can combine them in different ways to form distinct numbers.First, let's list all the possible factors of 441. Since 441 = 1 * 441, 3 * 147, 7 * 63, 9 * 49, 21 * 21. Wait, but factors come in pairs. But since we need four numbers, maybe we can break it down into more factors. Let's see.Alternatively, since we have to write 441 as a product of four distinct integers, let's try to find such combinations. Let's start with the smallest possible factors. The smallest distinct positive integers are 1, 2, 3, etc., but we have to make sure their product is 441.But wait, 441 is an odd number, so all factors must be odd. Therefore, 2 cannot be a factor. So the numbers a, b, c, d must all be odd. So possible numbers are 1, 3, 5, 7, 9, 15, 21, etc. Let me see.But 441 is 3² * 7². So all the factors must be products of these primes. So possible factors are 1, 3, 7, 9, 21, 49, 63, 147, 441. Wait, let me list all factors:1, 3, 7, 9, 21, 49, 63, 147, 441. These are the factors of 441. So there are nine factors in total. But we need four distinct factors. Hmm, but the product of four of them has to be 441. Wait, but 441 is the product of all four numbers. So, if we take four factors from the list above, their product must be 441. But since 441 is the total product, the four numbers must multiply to 441. However, each of these numbers are factors, but their product might be more than 441 if they are larger. Wait, this seems confusing.Alternatively, maybe the four numbers a, b, c, d are such that each is a factor of 441, but since they have to be distinct and multiply to 441, perhaps they are formed by combining the prime factors in different ways. Let's think about that.We have 3² * 7². To split this into four distinct numbers, each of which is a product of some combination of these primes. But since the primes are 3 and 7, each number can be represented as 3^x * 7^y, where x is 0, 1, or 2 and y is 0, 1, or 2. However, since we need four distinct numbers, each with different combinations of x and y. Also, when multiplied together, the exponents of 3 should add up to 2 and the exponents of 7 should add up to 2.Let me try to list possible exponents for each number. For each number, (x_i, y_i) such that sum of x_i = 2 and sum of y_i = 2. Each number must have x_i and y_i such that they don't repeat. Wait, but the numbers can have the same exponents but different if the multiplication is different? Wait, no, the numbers have to be distinct. So for example, 3^1*7^0 = 3, 3^0*7^1 =7, 3^1*7^1=21, etc. So to get four distinct numbers, each with different combinations of exponents.Wait, let's try. Let's think of possible exponents:Number 1: (0,0) = 1Number 2: (1,0) = 3Number 3: (0,1) = 7Number 4: (1,1) = 21But then, multiplying them all together: 1*3*7*21 = 441. Yes! That works. Let's check: 1*3 is 3, 3*7 is 21, 21*21 is 441. Yes. So the numbers would be 1, 3, 7, 21. Their sum is 1 + 3 + 7 + 21 = 32. Wait, 32 is option B. But let's check the options again. The options are A:30, B:32, C:31, D:36. So 32 is an option. But let's see if there's another combination that might result in a smaller sum.Alternatively, maybe we can use other exponents. Let's see. Let's try:Number1: (2,0) = 9Number2: (0,2) =49Number3: (0,0) =1Number4: (0,0)=1, but wait, 1 is already used, and they need to be distinct. So that doesn't work. Alternatively:Number1: (2,0)=9Number2: (0,1)=7Number3: (0,1)=7, but duplicates. Not allowed.Alternatively, Number1: (2,1)= 3^2 *7^1=9*7=63Number2: (0,1)=7, but then 63 and 7 are factors, but we need four numbers. Let's see:If we have 63, then 441 divided by 63 is 7, so the remaining three numbers need to multiply to 7. But 7 is prime, so the remaining three numbers would have to be 1, 1, 7, but duplicates again. Not allowed. Hmm.Alternatively, let's think of another combination. What if we use 1, 3, 7, 21. That gives sum 32. Let's check if that's the minimal possible. Or maybe we can use different factors.Wait, let's list all possible combinations. The factors of 441 are 1, 3, 7, 9, 21, 49, 63, 147, 441. So possible four distinct factors whose product is 441. Let's try different combinations.First, the obvious one: 1, 3, 7, 21. Product:1*3*7*21=441, sum=32. Another combination: 1, 3, 9, something. Let's see: 1*3*9=27. Then 441 divided by 27 is 16.333... Not an integer. So 16.333 is not a factor. So that doesn't work. How about 1, 3, 49. Then 1*3*49=147, 441 divided by 147 is 3, but 3 is already in the list, so duplicates. Not allowed.Another combination: 1,7,9, something. 1*7*9=63, 441/63=7. So 7 is already there. So duplicates again.How about 3,7,9, something. 3*7*9=189, 441/189= 2.333... Not integer. Nope.Wait, 1, 9, 21, something. 1*9*21=189, 441/189=2.333... Not integer.Alternatively, 3, 7, 21, something. 3*7*21=441, so 441 divided by that is 1. So the numbers would be 1,3,7,21 which is the same as before.Another approach: Let's see if we can find four distinct factors other than 1,3,7,21. For example, could we have 3, 7, 1, 21? That's same as before. What about 1, 3, 49, 3? No, duplicates. 1,7, 9, 7? Duplicate 7s. Not allowed.Wait, perhaps another combination: 1, 3, 7, 21 is the only possible combination? Let's check.Alternatively, maybe using 3^2 as 9 and 7^2 as 49. So 9 and 49. Then the remaining two numbers would have to multiply to 441/(9*49) = 441/441=1. So 1 and 1, but duplicates. Not allowed. So that doesn't work.Alternatively, using 3^1, 7^1, 3^1, 7^1. That is 3,7,3,7 but duplicates again. So that's invalid.Alternatively, maybe we can split the exponents differently. For example:First number: 3^1*7^0=3Second number: 3^1*7^0=3 (duplicate, not allowed)Third number: 3^0*7^1=7Fourth number: 3^0*7^1=7 (duplicate). So that doesn't work.Alternatively:First number: 3^2*7^0=9Second number:3^0*7^2=49Third number:1Fourth number:1, but duplicates. Not allowed.Alternatively:First number:3^1*7^1=21Second number:3^1*7^0=3Third number:3^0*7^1=7Fourth number:1. Then 21*3*7*1=441. That's the same as before. Sum is 21+3+7+1=32. So same result.Is there any other way? Let's see. Maybe using 1, 3, 3, 49. But duplicates. 1,3,7,21 is the only possible combination without duplicates. Wait, but let's check factors again.Wait, factors of 441 include 1,3,7,9,21,49,63,147,441. So perhaps 1,3,7,21 is the only quartet. Wait, let's check 1,3,9, and something. 1*3*9=27, 441/27=16.333. Not an integer. 1,3,21, something. 1*3*21=63, 441/63=7. So 1,3,21,7. Which is same as before. Similarly, 1,7,9, something. 1*7*9=63, 441/63=7. So 1,7,9,7. Duplicate. So that's invalid.Alternatively, 3,7,9, something. 3*7*9=189, 441/189=2.333. Not integer. 3,7,21, something. 3*7*21=441, so something is 1. So 3,7,21,1. Again same as before.What about using higher numbers? Let's try. 1, 49, 3, something. 1*49*3=147, 441/147=3. So duplicates again. 1,49,7, something. 1*49*7=343, 441/343≈1.285. Not integer.Alternatively, 3,49, something. 3*49=147, 441/147=3. So duplicates. 7,9, something. 7*9=63, 441/63=7. So duplicates. 21,21, but duplicates. 63 and 7. 63*7=441, but then the other two numbers would have to be 1 and 1. Duplicates.Hmm. It seems like the only possible combination is 1,3,7,21. Therefore, sum is 1+3+7+21=32, which is option B. But wait, the options include 31,30,36. Let me double-check if there's another possible combination.Wait, maybe using numbers that are not all factors of 441? But the problem states that a, b, c, d are positive integers, so their product is 441, so each of them must divide 441. Because if abcd=441, then each of a,b,c,d divides 441. Because 441 divided by a is bcd, which is an integer, so a divides 441, similarly for others. Therefore, all four numbers must be factors of 441, which are 1,3,7,9,21,49,63,147,441. So no other numbers can be used. So we have to pick four distinct numbers from this list whose product is 441. The only combination that works is 1,3,7,21. Therefore, sum is 32. So the answer is B.But wait, let me check if 3, 3, 7, 7 can be used. But they are duplicates. So no. What about 1,9,49, and something? 1*9*49=441, so the fourth number would have to be 1. But 1 is already used. So duplicates.Alternatively, 3,7,21, and something. 3*7*21=441, so the fourth number is 1. So same as before.Therefore, I think the answer is 32, which is option B. But let me check the options again. The problem says the options are A:30, B:32, C:31, D:36. So B is 32, which is the sum we got. However, I need to be careful here because sometimes there can be another combination. Let's verify once more.Is there a way to get a sum of 31? 31 is one less than 32. Maybe by replacing one of the numbers with a different combination. Let's see.Suppose we use 1,3,7,21 sum to 32. If we can find another set of four distinct factors with a sum of 31. Let's see. 31 is 1+3+7+20, but 20 is not a factor. 1+3+9+18=31, but 18 is not a factor. 1+5+... but 5 is not a factor. Wait, all factors are 1,3,7,9,21,49, etc. So let's see:Possible factors: 1,3,7,9,21,49. Let's check combinations of four numbers:1,3,7,21: sum 321,3,9, 441/(1*3*9)=441/27=16.333. Not integer.1,3,49: 1*3*49=147, 441/147=3. Duplicate.1,7,9: 1*7*9=63, 441/63=7. Duplicate.3,7,9: 3*7*9=189, 441/189=2.333. Not integer.1,21,3: same as before. 1,3,7,21.Alternatively, 1,3,49,3: duplicates. 1,7,21,3: same as before.Another way: Maybe split the exponents differently. For example, if we take 3^1, 3^1, 7^1, 7^1, but those are duplicates, so not allowed. Alternatively, using 3^2 and 7^1. 9 and 7. Then the remaining two numbers would need to multiply to 441/(9*7)=441/63=7. So 7 and 1, but 7 is already used. So duplicates.Alternatively, 9 and 49. 9*49=441, then remaining two numbers are 1 and 1. Duplicates.Alternatively, 21 and 21. Duplicates. 3 and 147. 3*147=441, remaining two numbers 1 and 1. Duplicates.Alternatively, 63 and 7. 63*7=441. Remaining numbers 1 and 1. Duplicates.Hmm. It seems like the only valid combination is 1,3,7,21. So sum is 32. Therefore, the answer is B. But let me check if there is another way. Wait, what if we use 1, 1, 21, 21. But duplicates again. Not allowed. What about 3, 3, 7,7. Duplicates. So no.Alternatively, using the number 9. If we use 9, then we have to see if the remaining three numbers can multiply to 441/9=49. So 49 can be factored as 1,7,7. But duplicates. So that's invalid. Or 49= 1,1,49. Duplicates again.Alternatively, 9 and 49. Then 9*49=441. So remaining numbers 1 and 1. Duplicates.Alternatively, 3, 3, 7, 7: duplicates.So no. Therefore, the only possible combination is 1,3,7,21. Sum is 32. Therefore, the answer is B.But wait, just to ensure, maybe there's a way to use the factor 49. Let's see. If I take 49, then the remaining three numbers have to multiply to 441/49=9. So 9 can be factored into 1,3,3. But duplicates. So 1,3,3,49. Duplicate 3s. Not allowed. Alternatively, 9=1,1,9. Then numbers 1,1,9,49. Duplicates again. So invalid.Similarly, using 21. If we take 21, then remaining numbers have to multiply to 441/21=21. Then 21 can be 1,3,7. So numbers 1,3,7,21. Same as before.Alternatively, 21 can be 3,7,1. Same thing. So yes, that's the only combination. Therefore, 32 is the sum, so B is the answer.Final Answerboxed{B}</think>Given that (a), (b), (c), and (d) are distinct positive integers, and their product (abcd = 441), we need to find the value of (a + b + c + d).First, we factorize 441. The prime factorization of 441 is (3^2 times 7^2). We need to distribute these prime factors into four distinct positive integers. The factors of 441 are: 1, 3, 7, 9, 21, 49, 63, 147, 441. We need four distinct factors whose product is 441. By examining the possible combinations, we find that the factors 1, 3, 7, and 21 multiply to 441:[1 times 3 times 7 times 21 = 441]The sum of these factors is:[1 + 3 + 7 + 21 = 32]After verifying all possible combinations and ensuring there are no duplicates, we conclude that the only valid combination is 1, 3, 7, and 21. Therefore, the value of (a + b + c + d) is (boxed{B}).,<think>Okay, let's see. The problem says that a, b, c, d are distinct positive integers, and their product abcd is 441. We need to find the value of a + b + c + d, and the options are 30, 32, 31, 36. Hmm.First, I need to factorize 441 to find possible integers a, b, c, d. Let me recall that 441 is 21 squared, so 21*21, but since they have to be distinct, I can't just use 1, 3, 7, 21 because 1*3*7*21 = 441, but let me check that. 1*3 is 3, 3*7 is 21, 21*21 is 441. Wait, but 21 is repeated here. So that's not allowed because they have to be distinct. So maybe I need to break down 441 further into its prime factors. Let's do that.441 divided by 3 is 147. 147 divided by 3 is 49. 49 is 7 squared. So the prime factors of 441 are 3*3*7*7. So that's 3² * 7². Now, we need to split these prime factors into four distinct positive integers. Since the prime factors are only 3s and 7s, we need to combine them in different ways to get four distinct integers.Let me think. The prime factors are 3, 3, 7, 7. To make four distinct integers, each of them needs to be a product of some combination of these primes. Let's list possible factors of 441:1, 3, 7, 9, 21, 49, 63, 147, 441.But since we need four numbers, and their product is 441, so all four numbers must be factors of 441. Also, they have to be distinct. Let's see.Possible factors: 1, 3, 7, 9, 21, 49, 63, 147, 441. But since the product is 441, and there are four numbers, none of them can be too large. For example, 441 is too big because 441*1*1*1 would be 441, but we need four numbers. Similarly, 147 is also too large because 147*3*1*1 is 441, but again, duplicates.So probably the numbers are smaller. Let's consider using 1, 3, 7, and 21. Wait, 1*3*7*21 = 441, but 21 is 3*7. However, all numbers are distinct here. Wait, 1, 3, 7, 21. Let's check the product: 1*3=3, 3*7=21, 21*21=441. Wait, no, 1*3*7*21 is indeed 441. Let me compute that again: 1*3=3, 3*7=21, 21*21=441. Wait, no, that's 1*3*7*21. So 1 times 3 is 3, times 7 is 21, times 21 is 441. Yes, that works. And all numbers are distinct: 1, 3, 7, 21. Then their sum is 1+3+7+21=32. Hmm, 32 is one of the options, option B. But let me check if there's another combination that gives a smaller sum. Maybe we can find a combination with smaller numbers?Wait, but 1 is the smallest positive integer, so we can't get smaller than that. Let's see if there's another set. For example, can we use 1, 3, 7, and 21? That's the one we have. Alternatively, maybe 1, 7, 9, and something else. Let's try. 1*7*9*x = 441. So x = 441 / (1*7*9) = 441 / 63 = 7. But 7 is already in the set, so duplicates. Not allowed.What about 1, 3, 9, something? 1*3*9*x=441, so x=441/(27)=16.333… Not an integer. So that's no good.How about 3, 7, 9, and something? 3*7*9=189, so 441/189=2.333… Not an integer. Hmm.Alternatively, 1, 3, 7, 21 seems to be the only combination. Let's check another possibility. Maybe 3, 7, 21, and 1. That's the same as above. What if we use 1, 9, 7, and 7? But 7 is repeated. Not allowed.Wait, maybe 1, 3, 21, 7. Same as before. Alternatively, 3, 7, 1, 21. Still the same. What if we try to avoid 1? Let's see. If we don't use 1, then the numbers would have to be at least 3, 7, 9, something. Let's check: 3*7*9*x=441, so x=441/(189)=2.333… Not integer. Not valid. If we try 3, 7, 3, 7. But duplicates again. So, seems like we have to include 1.Alternatively, can we get four numbers without 1? Let me see. Let's factor 441 into four numbers. 441=3²*7². To split into four distinct factors, each factor must be a combination of 3s and 7s.Possible factors: 3^0*7^0=1, 3^1*7^0=3, 3^0*7^1=7, 3^2*7^0=9, 3^0*7^2=49, 3^1*7^1=21, 3^2*7^1=63, 3^1*7^2=147, 3^2*7^2=441.So, possible factors are 1,3,7,9,21,49,63,147,441.We need four distinct factors. Let's try to pick four distinct ones whose product is 441. Let's think of combinations. Let's try 1,3,7,21. Product is 1*3*7*21=441. Sum is 1+3+7+21=32. That's option B.Is there another combination? Let's see. What if we use 1,3,9, and then 441/(1*3*9)=441/27=16.333… Not integer. So invalid. How about 1,7,9, and 441/(1*7*9)=441/63=7. But 7 is already in there. Duplicate. Not allowed.What about 3,7,9, and 441/(3*7*9)=441/189=2.333… Not integer. Hmm.How about 1,3,49, and 441/(1*3*49)=441/147=3. So 1,3,49,3. But duplicate 3. Not allowed.Wait, 1,3,21,7: already considered. How about 3,7,21,1. Same thing. Hmm.Another approach: Since 441=3²*7², the number of divisors is (2+1)(2+1)=9. So there are 9 divisors as we listed. To split into four distinct factors, perhaps we can use exponents. Let's see. Each factor is 3^a *7^b, where a can be 0,1,2 and b can be 0,1,2. But the product of all four factors should be 3^2*7^2. So the sum of exponents of 3 in each factor should total 2, and same for 7.So, for each factor, the exponents of 3 and 7 must be such that:For 3's exponents: a1 + a2 + a3 + a4 = 2For 7's exponents: b1 + b2 + b3 + b4 = 2Each ai and bi are non-negative integers, and each factor 3^{ai}7^{bi} must be distinct.So we need to assign exponents such that each (ai, bi) is unique, and sum ai=2, sum bi=2.Possible assignments:For ai: each ai can be 0,1, or 2, but sum to 2. So possible combinations for ai:0,0,0,20,0,1,1Similarly for bi: same.But since all factors must be distinct, we need to have different combinations of ai and bi.Let me try to find four distinct factors where exponents of 3 sum to 2 and exponents of 7 sum to 2.Let's start with ai:Option 1: two 1s and two 0s. For example: 0,0,1,1. Sum is 2.Similarly for bi: 0,0,1,1. Then the factors would be:For ai=0, bi=0: 1ai=0, bi=1: 7ai=1, bi=0:3ai=1, bi=1:21So the factors are 1,3,7,21. Which is exactly the set we had before, sum 32. So that's one possibility.Another possibility for ai: 0,0,0,2. So three 0s and one 2. Similarly for bi: 0,0,0,2. Then the factors would be:For ai=0, bi=0:1ai=0, bi=2:49ai=2, bi=0:9ai=0, bi=0:1 (duplicate). So that's a problem. Because ai=0, bi=0 is 1, but we already have that. So duplicates. So that doesn't work.Alternatively, ai could be 0,0,0,2 and bi as 0,0,1,1. Then factors:ai=0, bi=0:1ai=0, bi=1:7ai=0, bi=1:7 (duplicate)ai=2, bi=0:9So duplicates again. So that doesn't work.Alternatively, ai=0,0,1,1 and bi=0,0,0,2.Then factors:ai=0, bi=0:1ai=0, bi=0:1 (duplicate)ai=1, bi=0:3ai=1, bi=2: 3*49=147But duplicate 1. Not allowed.Hmm. So seems like the only valid combination without duplicates is the first one: ai=0,0,1,1 and bi=0,0,1,1, giving factors 1,3,7,21.So the sum is 1+3+7+21=32, which is option B. But let me check the answer options again. The options are A:30, B:32, C:31, D:36. So B is 32. But wait, is there another combination? Let me check again.Alternatively, maybe using 9 instead of 21? Wait, if we use 9, then let's see:Suppose the factors are 1,3,7,9. Product is 1*3*7*9=189. Not 441. Not enough. So we need a bigger number. So 21 is needed.Alternatively, 3,7, 3,7. But duplicates. Not allowed. Hmm.Wait, another thought. Maybe using 1, 3, 7, 21 is the only possible combination. Let me check all possible sets of four distinct factors of 441.List of factors:1,3,7,9,21,49,63,147,441.Trying to pick four distinct numbers from these whose product is 441.Let's check:1,3,7,21: product=441, sum=32.1,3, 49, 3: duplicate 3. Not allowed.1,7, 9, 7: duplicate 7. Not allowed.3,7,21,1: same as above.1,9, 49, 1: duplicates. Not allowed.3,7, 9, something: 3*7*9=189. 441/189=2.333… Not integer.How about 1, 21, 21, 1: duplicates. Not allowed.Another combination: 3, 3, 7,7: duplicates. No.How about 1, 3, 49, and 3: duplicates again.Alternatively, 3,7,21,1: same as first case.Wait, another idea. What if we use 1, 7, 9, and 7? But duplicates. No.Wait, perhaps using 9, 49, but 9*49=441. So if we have 1, 9, 49, and 1. But duplicates again. So no.Alternatively, 3, 3, 49, 49: duplicates. Not allowed.Wait, maybe 3, 7, 21, 1. That's the same as before.Alternatively, 7, 9, something. 7*9=63. 441/63=7. So 7,9,7, something. Duplicate again.Alternatively, 3, 21, 7,1. Same as before.Alternatively, 9, 21, and two ones. But duplicates.Hmm. So it seems like the only possible combination of four distinct positive integers multiplying to 441 is 1,3,7,21, giving a sum of 32, which is option B. But let me check once more if there's another combination.Wait, 441 divided by 1*3*7 is 21, which we have. What if we use 3,7, 21,1. That's same. Alternatively, if we use 1, 3, 49, and 3: duplicates. 1, 7, 21, 3: same as before. 1,9, 49, 7: product is 1*9*49*7= 9*343=3087. No, that's way too big. Wait, 9*49=441, so 1*7*9*7=441, but duplicates. Not allowed.Wait, another way: 3*3*7*7=441, but those are all duplicates. So no. So 1,3,7,21 is the only possible set. Therefore, the answer should be 32, which is option B. But let me check the options again. The question says the options are A:30, B:32, C:31, D:36. So B is 32. But the answer given in the options is B? Wait, but I need to confirm once more.Wait, let me check 1,3,7,21. Their product: 1*3=3, 3*7=21, 21*21=441. Yes, correct. Sum:1+3+7+21=32. So that's correct. So unless there's another combination, but I don't see any. Therefore, the answer is 32, option B.Wait, but before I finalize, let me consider if there's another combination where numbers aren't necessarily including 1 but still distinct. Let's try 3, 7, 21, and 1, which we've already considered. What about 3, 7, 9, something? 3*7*9=189. 441/189=2.333, which isn't integer. So no.How about 3,7, 3, 7: duplicates. No.Alternatively, 7, 21, 3, 1: same as before.Alternatively, 1, 9, 49, and 1: duplicates. Not allowed.Alternatively, 9, 49, 3, and something. 9*49=441. So 441/(9*49)=1. So 1,9,49, but then we need a fourth number. That would be 1,9,49,1: duplicates. Not allowed.Alternatively, 3, 9, 7, and something. 3*9*7=189. 441/189=2.333. Not integer. So no.Alternatively, 3, 21, 7,1: same as before.Therefore, there seems to be no other combination. Hence, the answer should be B:32.Final Answerboxed{B}</think>Given that (a), (b), (c), and (d) are distinct positive integers, and their product (abcd = 441), we need to find the value of (a + b + c + d).First, we factorize 441 into its prime factors:[441 = 3^2 times 7^2]We need to split these prime factors into four distinct positive integers. The factors of 441 are: 1, 3, 7, 9, 21, 49, 63, 147, 441. We need to find four distinct factors whose product is 441.After considering various combinations, we find that the factors 1, 3, 7, and 21 multiply to 441:[1 times 3 times 7 times 21 = 441]These factors are distinct, and their sum is:[1 + 3 + 7 + 21 = 32]We verify that there are no other combinations of four distinct factors of 441 that result in a different sum. Therefore, the value of (a + b + c + d) is (boxed{B}).]

🔑:Since 441 = 3 times 3 times 7 times 7 = 1 times 3 times 7 times 21, it follows that abcd = 1 times 3 times 7 times 21, thus, corresponding to a, b, c, d are 1, 3, 7, 21 respectively, a+b+c+d = 1+3+7+21 = 32, therefore, the correct choice is boxed{B}.

🧠:[<think>Alright, let's tackle these complex number operations step by step. I remember that when dealing with complex numbers, the key is to treat (i) as the imaginary unit where (i^2 = -1). Let's start with part a) ((2 + 3i)^2).First, I need to recall how to square a binomial. The formula is ((a + b)^2 = a^2 + 2ab + b^2). Applying this to (2 + 3i), where (a = 2) and (b = 3i):[(2 + 3i)^2 = 2^2 + 2 times 2 times 3i + (3i)^2]Calculating each term:1. (2^2 = 4)2. (2 times 2 times 3i = 12i)3. ((3i)^2 = 9i^2)Now, since (i^2 = -1), the third term becomes (9 times (-1) = -9). Putting all the terms together:[4 + 12i - 9 = (4 - 9) + 12i = -5 + 12i]So, part a) is (-5 + 12i). Let me double-check my steps. Squared the first term, doubled the product, squared the second term, replaced (i^2) with (-1). Looks good.Moving on to part b) ((3 - 5i)^2). Again, using the binomial square formula ((a - b)^2 = a^2 - 2ab + b^2). Here, (a = 3) and (b = 5i):[(3 - 5i)^2 = 3^2 - 2 times 3 times 5i + (5i)^2]Calculating each term:1. (3^2 = 9)2. (2 times 3 times 5i = 30i), so subtracting that gives (-30i)3. ((5i)^2 = 25i^2 = 25 times (-1) = -25)Combining the terms:[9 - 30i - 25 = (9 - 25) - 30i = -16 - 30i]So part b) is (-16 - 30i). Checking again: squared terms, middle term with the negative sign, and substitution of (i^2). Seems correct.Now part c) ((5 + 3i)^3). This is a cube, so the formula is a bit more involved. The formula for ((a + b)^3 is (a^3 + 3a^2b + 3ab^2 + b^3). Let's apply this with (a = 5) and (b = 3i):[(5 + 3i)^3 = 5^3 + 3 times 5^2 times 3i + 3 times 5 times (3i)^2 + (3i)^3]Let's compute each term step by step:1. (5^3 = 125)2. (3 times 5^2 times 3i = 3 times 25 times 3i = 225i)3. (3 times 5 times (3i)^2). First, compute ((3i)^2 = 9i^2 = -9). Then multiply: (3 times 5 times (-9) = 15 times (-9) = -135)4. ((3i)^3 = 27i^3). Now, (i^3 = i^2 times i = -1 times i = -i), so (27i^3 = 27(-i) = -27i)Now, combining all these terms:[125 + 225i - 135 - 27i]Combine like terms:Real parts: (125 - 135 = -10)Imaginary parts: (225i - 27i = 198i)So, putting it all together:[-10 + 198i]Wait a second, let me check the third term again. The term (3 times 5 times (3i)^2): (3i) squared is (9i^2 = -9), then (3 times 5 times (-9)) is indeed (15 times (-9) = -135). And the fourth term: ( (3i)^3 = 27i^3 = 27(-i) = -27i). Then combining:125 + 225i -135 -27i. Real parts 125-135 is -10, imaginary parts 225i-27i is 198i. Yep, that's correct.So part c) is (-10 + 198i). Hmm, but let me verify using another method to be sure. Sometimes when expanding cubes, it's easy to make a mistake in the coefficients.Alternatively, I can compute ((5 + 3i)^3) by first computing ((5 + 3i)^2) and then multiplying by (5 + 3i) again.First, compute ((5 + 3i)^2):Using ((a + b)^2 = a^2 + 2ab + b^2):(5^2 + 2 times 5 times 3i + (3i)^2 = 25 + 30i + 9i^2 = 25 + 30i -9 = 16 + 30i)Then multiply this result by (5 + 3i):[(16 + 30i)(5 + 3i)]Using the distributive property (FOIL):First: (16 times 5 = 80)Outer: (16 times 3i = 48i)Inner: (30i times 5 = 150i)Last: (30i times 3i = 90i^2 = 90(-1) = -90)Combine all terms:80 + 48i + 150i -90 = (80 - 90) + (48i + 150i) = -10 + 198iSame result! So that confirms part c) is indeed (-10 + 198i).Let me recap all parts:a) After expanding, we got (-5 + 12i)b) After expanding, we got (-16 - 30i)c) After expanding both directly and by squaring then multiplying, we got (-10 + 198i)I think that's all correct. Let me just check for any arithmetic errors.For part a):Original expansion: (4 + 12i -9 = -5 +12i). Correct.Part b):(9 -30i -25 = -16 -30i). Correct.Part c):First method: 125 +225i -135 -27i = -10 +198i. Second method: (16 +30i)(5+3i) = -10 +198i. Both methods same result. So yes, correct.Therefore, all answers are correct.Final Answera) boxed{-5 + 12i} b) boxed{-16 - 30i} c) boxed{-10 + 198i}</think>To solve the given problems involving complex numbers, we use algebraic expansion and the property (i^2 = -1). Part a) ((2 + 3i)^2)Using the binomial expansion formula ((a + b)^2 = a^2 + 2ab + b^2):[(2 + 3i)^2 = 2^2 + 2 cdot 2 cdot 3i + (3i)^2][= 4 + 12i + 9i^2][= 4 + 12i - 9 quad (text{since } i^2 = -1)][= -5 + 12i] Part b) ((3 - 5i)^2)Using the binomial expansion formula ((a - b)^2 = a^2 - 2ab + b^2):[(3 - 5i)^2 = 3^2 - 2 cdot 3 cdot 5i + (5i)^2][= 9 - 30i + 25i^2][= 9 - 30i - 25 quad (text{since } i^2 = -1)][= -16 - 30i] Part c) ((5 + 3i)^3)First, compute ((5 + 3i)^2) using the binomial expansion formula:[(5 + 3i)^2 = 5^2 + 2 cdot 5 cdot 3i + (3i)^2][= 25 + 30i + 9i^2][= 25 + 30i - 9 quad (text{since } i^2 = -1)][= 16 + 30i]Then multiply the result by (5 + 3i):[(16 + 30i)(5 + 3i)]Using the distributive property:[= 16 cdot 5 + 16 cdot 3i + 30i cdot 5 + 30i cdot 3i][= 80 + 48i + 150i + 90i^2][= 80 + 198i - 90 quad (text{since } i^2 = -1)][= -10 + 198i] Final Answersa) (boxed{-5 + 12i}) b) (boxed{-16 - 30i}) c) (boxed{-10 + 198i}),<think>Alright, let's tackle these complex number operations one by one. I remember that when you square or cube a complex number, you have to use the distributive property, just like with binomials. Let me start with part a: (2 + 3i) squared. Okay, so (a + b)^2 is a^2 + 2ab + b^2. Applying that here, a is 2 and b is 3i. So first, square 2: that's 4. Then, 2 times 2 times 3i: 2*2*3i is 12i. Then, square 3i: (3i)^2. Hmm, 3 squared is 9, and i squared is -1, so that's 9*(-1) = -9. So putting it all together: 4 + 12i - 9. Combine the real parts: 4 - 9 is -5. So the result is -5 + 12i. Let me double-check that. Multiply (2 + 3i)(2 + 3i): 2*2 is 4, 2*3i is 6i, 3i*2 is 6i, and 3i*3i is 9i^2. So 4 + 6i + 6i + 9(-1) = 4 + 12i -9 = same as before, -5 +12i. Yep, that's correct. Moving on to part b: (3 - 5i)^2. Again, using the same formula (a - b)^2 = a^2 - 2ab + b^2. Here, a is 3 and b is 5i. First, 3 squared is 9. Then, 2*3*5i: that's 30i. But since it's minus, it'll be -30i. Then, (5i)^2: 25i^2, which is 25*(-1) = -25. So combining all the parts: 9 - 30i -25. Combine the real numbers: 9 -25 is -16. So the result is -16 -30i. Let me verify by expanding: (3 -5i)(3 -5i). 3*3=9, 3*(-5i)= -15i, (-5i)*3= -15i, (-5i)*(-5i)=25i^2. So total is 9 -15i -15i +25(-1) = 9 -30i -25 = -16 -30i. Correct. Now part c is a bit more complicated: (5 + 3i)^3. Cubing a binomial. The formula for (a + b)^3 is a^3 + 3a^2b + 3ab^2 + b^3. Let's apply that here. a is 5, b is 3i. First term: a^3 = 5^3 = 125. Second term: 3a^2b = 3*(5^2)*(3i) = 3*25*3i = 225i. Third term: 3ab^2 = 3*5*(3i)^2. Let's compute (3i)^2 first: 9i^2 = -9. So 3*5*(-9) = 15*(-9) = -135. Fourth term: b^3 = (3i)^3. That's 27i^3. But i^3 is i^2*i = (-1)*i = -i. So 27*(-i) = -27i. Now combine all the terms: 125 + 225i -135 -27i. Combine real parts: 125 -135 = -10. Combine imaginary parts: 225i -27i = 198i. So the result is -10 +198i. Wait, let me check by expanding step by step. First, compute (5 +3i)^2 first, then multiply by (5 +3i) again. (5 +3i)^2: Let's do this. 5^2 =25, 2*5*3i=30i, (3i)^2=9i^2=-9. So 25 +30i -9=16 +30i. Then multiply that by (5 +3i): (16 +30i)(5 +3i). Multiply 16*5=80, 16*3i=48i, 30i*5=150i, 30i*3i=90i^2=90*(-1)=-90. Combine terms: 80 +48i +150i -90. Real parts: 80 -90= -10. Imaginary parts: 48i +150i=198i. So result is -10 +198i. That matches. Therefore, the answers are:a) -5 +12ib) -16 -30ic) -10 +198iI think that's all. Let me just make sure I didn't mix up any signs. For part a, squaring 2+3i, everything looked good. Part b, subtracting 5i, so the cross term is negative, and the squared term is negative as well. Part c, even though it's cubed, breaking it down step by step worked out. Yeah, seems correct.Final Answera) boxed{-5 + 12i}b) boxed{-16 - 30i}c) boxed{-10 + 198i}</think>To solve the given problems, we use the distributive property (FOIL method) for complex numbers and binomial expansion for cubing. Part a: ((2 + 3i)^2)Using the binomial expansion ((a + b)^2 = a^2 + 2ab + b^2):[(2 + 3i)^2 = 2^2 + 2 cdot 2 cdot 3i + (3i)^2][= 4 + 12i + 9i^2]Since (i^2 = -1):[= 4 + 12i - 9][= -5 + 12i] Part b: ((3 - 5i)^2)Using the binomial expansion ((a - b)^2 = a^2 - 2ab + b^2):[(3 - 5i)^2 = 3^2 - 2 cdot 3 cdot 5i + (5i)^2][= 9 - 30i + 25i^2]Since (i^2 = -1):[= 9 - 30i - 25][= -16 - 30i] Part c: ((5 + 3i)^3)Using the binomial expansion ((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3):[(5 + 3i)^3 = 5^3 + 3 cdot 5^2 cdot 3i + 3 cdot 5 cdot (3i)^2 + (3i)^3]Calculating each term:[5^3 = 125][3 cdot 5^2 cdot 3i = 3 cdot 25 cdot 3i = 225i][3 cdot 5 cdot (3i)^2 = 3 cdot 5 cdot 9i^2 = 135 cdot (-1) = -135][(3i)^3 = 27i^3 = 27 cdot (-i) = -27i]Combining all terms:[125 + 225i - 135 - 27i][= (125 - 135) + (225i - 27i)][= -10 + 198i] Final Answersa) (boxed{-5 + 12i})b) (boxed{-16 - 30i})c) (boxed{-10 + 198i})]

🔑:Solution. quad a) quad(2+3 i)^{2}=4+2 cdot 2 cdot 3 i+9 i^{2}=4+12 i-9=-5+ +12 ib) (3-5 i)^{2}=9-2 cdot 3 cdot 5 i+25 i^{2}=9-30 i-25=-16-30 i;c) (5+3 i)^{3}=125+3 cdot 25 cdot 3 i+3 cdot 5 cdot 9 i^{2}+27 i^{3};since i^{2}=-1, and i^{3}=-i, we get (5+3 i)^{3}=125+225 i-135- -27 i=-10+198 i.183-190. Perform the operations:

🧠:[<think>Okay, so I need to figure out what 3 plus negative 2 is. Hmm, let's start by recalling how addition works with positive and negative numbers. I remember that adding a negative number is like subtracting its absolute value. So, maybe 3 + (-2) is the same as 3 minus 2? Let me check that.If I have 3 apples and someone takes away 2 apples, how many do I have left? That would be 1 apple. So, 3 minus 2 equals 1. But wait, the original problem was 3 plus (-2). Does that mean the same thing? Yeah, because adding a negative number is equivalent to subtracting a positive number. So, yes, 3 + (-2) should be 1.Let me visualize it on a number line. Starting at 3, if I add a negative 2, I move 2 units to the left. So, from 3, moving left 2 units lands me on 1. That makes sense. Alternatively, if I think of it as debt, if I have 3 and I owe someone 2, then my net amount is 1.Another way to think about it is using the rules of signs. When the signs are different, we subtract the smaller number from the larger one and take the sign of the larger number. Here, 3 is positive and 2 is negative. The larger absolute value is 3, so subtract 2 from 3, which is 1, and keep the positive sign. So, the result is 1.Wait, just to make sure I didn't mix up any rules. Let me verify with another example. If it were 3 + 2, that's 5. If it's 3 + (-2), that's like 3 - 2, which is 1. If it were -3 + 2, then it's -1, right? Because you subtract 2 from 3 and take the sign of the larger absolute value, which is negative. So, yes, seems consistent.So, putting it all together, 3 plus negative 2 equals 1. All the different ways I thought about it lead to the same answer, so I'm pretty confident that's correct.Final AnswerThe result of 3 + (-2) is boxed{1}.</think>To calculate (3 + (-2)):1. Recognize that adding a negative number is equivalent to subtracting its absolute value.2. Therefore, (3 + (-2)) is the same as (3 - 2).3. Subtracting 2 from 3 gives the result 1.4. Verify using different methods (number line, debt analogy, sign rules) all lead to the same result.The result of (3 + (-2)) is (boxed{1}).,<think>Okay, so I need to figure out what 3 plus negative 2 is. Hmm, let's start by recalling how addition works with positive and negative numbers. I remember that adding a negative number is like subtracting its absolute value. So, 3 + (-2) should be the same as 3 minus 2. Let me check that again.If I have 3 apples and someone takes away 2 apples, how many apples do I have left? That's right, I would have 1 apple left. So maybe 3 + (-2) is 1? Wait, but is there another way to think about it to confirm?Another method could be using a number line. Starting at 3 on the number line, adding a negative number means moving to the left. So, if I start at 3 and move 2 units to the left, where do I land? Let's visualize: 3, then 2 steps left would be 2, then 1. Yep, that's 1. That matches the previous result.Alternatively, I can think about the rules of signs. When you add numbers with different signs, you subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value. Here, 3 has an absolute value of 3, and -2 has an absolute value of 2. Since 3 is larger than 2, we subtract 2 from 3, which gives 1, and the result is positive because 3 was positive. So again, that gives 1.Wait, but let me make sure I'm not mixing up any rules. What if both numbers were negative? Then you would add their absolute values and keep the negative sign. But here, one is positive and one is negative, so subtraction applies. Right, different signs mean subtract. So 3 + (-2) is 3 - 2, which is 1. I can also check this using algebra tiles or counters. If positive numbers are represented by, say, blue tiles and negative numbers by red tiles. So 3 blue tiles plus 2 red tiles. Each red tile cancels a blue tile. So 2 red tiles would cancel 2 blue tiles, leaving 1 blue tile. That's 1. Alternatively, using debt as an example. If I have 3 and I owe someone 2, my net amount is 1. So combining 3 and a debt of 2 gives me 1. That makes sense.I think all these different methods lead to the same answer: 1. So 3 + (-2) must equal 1. Let me just verify once more. If I do 3 + (-2), that's the same as 3 - 2. 3 minus 2 is definitely 1. Yep. So there's no doubt here. The answer has to be 1. I don't see any other possible result here. All the different approaches—number line, algebra tiles, debt analogy, and the arithmetic rules—agree. Therefore, I'm confident that the correct answer is 1.Final AnswerThe result of 3+(-2) is boxed{1}.</think>To calculate (3 + (-2)), we can use the rule that adding a negative number is equivalent to subtracting its absolute value. 1. Start with the expression (3 + (-2)).2. Recognize that adding (-2) is the same as subtracting 2 from 3.3. Perform the subtraction: (3 - 2 = 1).We verified this using multiple methods: number line visualization, algebraic rules, and real-life analogies (such as debt). All methods confirm that the result is 1.[3 + (-2) = 3 - 2 = 1]The result of (3 + (-2)) is (boxed{1}).]

🔑:To calculate 3+(-2), we follow these steps:1. Recognize that adding a negative number is the same as subtracting its absolute value: 3 + (-2).2. Perform the subtraction: 3 - 2.3. Calculate the result: 1.Therefore, the calculation simplifies as follows:3 + (-2) = 3 - 2 = 1.Thus, the final answer is boxed{1}.

🧠:[<think>Okay, let's try to solve this problem. So, we have 48 boys and 32 girls, and they need to be arranged in n rows (with n > 1). In each row, the number of boys and the number of girls must be equal. We need to find out how many different ways they can be arranged and how many boys and girls are in each row for each arrangement.Hmm, let me start by understanding the problem correctly. The key points are:1. We have two groups: 48 boys and 32 girls.2. They need to be arranged in n rows, where n is greater than 1.3. In each row, the number of boys and girls must be the same.Wait, does this mean that each row has the same number of boys and the same number of girls? So, for example, if there are x boys in each row, there must also be x girls in each row? Or does it mean that each row has the same total number of children, composed of boys and girls?Wait, the problem says "the number of boys and girls in each row being equal." So, maybe each row must have the same number of boys and the same number of girls. So, in each row, the count of boys is equal to the count of girls. Wait, but that would mean each row has the same number of boys as girls. For example, if each row has 2 boys and 2 girls, then the total number of boys would be 2n, and the total number of girls would be 2n. But here, we have 48 boys and 32 girls, so if each row has k boys and k girls, then total boys would be k*n and total girls would be k*n. But 48 ≠ 32, so that would be impossible. Therefore, maybe I misinterpret the problem.Wait, maybe the problem is that in each row, the number of boys is equal to the number of girls. That is, in each row, boys and girls are equal in number. So, each row has the same number of boys and girls. So, if each row has b boys and g girls, then b = g for each row. Then, total boys would be n*b, total girls would be n*g. But since b = g, then total boys and total girls would both be n*b. But since 48 ≠ 32, this can't happen. Therefore, this can't be the case. So perhaps my interpretation is wrong.Wait, let me check the original problem again: "the number of boys and girls in each row being equal." Maybe it's the number of boys and the number of girls in each row must be equal across all rows, but not necessarily within a single row. That is, each row has the same number of boys and each row has the same number of girls, but boys and girls per row can be different. For example, each row has 6 boys and 4 girls, so in each row, boys and girls might not be equal, but the number of boys per row is same across all rows, and the number of girls per row is same across all rows.Yes, that makes more sense. So, the problem is that in each row, the number of boys is the same for all rows, and the number of girls is the same for all rows. So, for example, if there are 6 boys per row and 4 girls per row, then with 8 rows, we'd have 48 boys and 32 girls. So, this would satisfy the problem. So, the number of boys per row times the number of rows equals 48, and the number of girls per row times the number of rows equals 32. So, if we let b be the number of boys per row, g be the number of girls per row, and n be the number of rows, then:b * n = 48g * n = 32And since n > 1, we need to find all possible integer values of n, b, g such that these equations hold. Then, the different arrangements would correspond to different divisors n of both 48 and 32, since n must divide both 48 and 32. Therefore, n must be a common divisor of 48 and 32. Since n > 1, we need to find all common divisors of 48 and 32 greater than 1.Wait, let me confirm this. So, if n is the number of rows, then n must divide both 48 and 32, so that 48/n and 32/n are integers (since b = 48/n and g = 32/n must be integers). Therefore, the number of different arrangements is equal to the number of common divisors of 48 and 32 that are greater than 1. For each such common divisor n, the number of boys per row would be 48/n, and the number of girls per row would be 32/n.Therefore, first, we need to compute the greatest common divisor (GCD) of 48 and 32, and then list all the divisors of that GCD, excluding 1.Let me compute GCD(48, 32). Let's do prime factorization.48: 2^4 * 3^132: 2^5The GCD is the minimum of each prime exponent, so for 2, min(4,5) = 4, and 3^0. So GCD is 2^4 = 16.Therefore, the common divisors of 48 and 32 are the divisors of 16. The divisors of 16 are: 1, 2, 4, 8, 16.But since n > 1, we exclude 1. Therefore, the possible values of n are 2, 4, 8, 16.Thus, there are 4 different ways to arrange them.For each n, we can compute the boys per row (48/n) and girls per row (32/n):- For n=2: 48/2=24 boys per row, 32/2=16 girls per row.- For n=4: 48/4=12 boys, 32/4=8 girls.- For n=8: 48/8=6 boys, 32/8=4 girls.- For n=16: 48/16=3 boys, 32/16=2 girls.Therefore, the different arrangements correspond to these values of n, and each gives the number of boys and girls per row as above.Therefore, the number of different ways is 4, and for each arrangement, the boys and girls per row are (24,16), (12,8), (6,4), (3,2).Wait, let me check if 16 rows is possible. If n=16, then each row has 3 boys and 2 girls, so 16 rows would require 48 boys and 32 girls, which matches the given numbers. So that works.Similarly, n=8: 8 rows of 6 boys and 4 girls: 8*6=48, 8*4=32. Correct.n=4: 4 rows of 12 boys and 8 girls: 4*12=48, 4*8=32. Correct.n=2: 2 rows of 24 boys and 16 girls: 2*24=48, 2*16=32. Correct.Therefore, the answer is that there are 4 different ways, and the boys and girls per row for each arrangement are as above.Wait, but let me check if there's a possibility that n can also be a factor of the GCD but the corresponding boys and girls per row might not be positive integers. But since GCD is 16, and n divides 16, and 48 and 32 are both divisible by 16, 8, 4, 2, etc., then each division would result in integer numbers of boys and girls per row. So, yes, the answer is 4 ways.Therefore, the number of different arrangements is 4, and for each arrangement, the number of boys and girls per row are (3,2), (6,4), (12,8), (24,16). Wait, but in the order of increasing n or decreasing? Wait, when n increases, the number of boys and girls per row decreases. So, when n=2, per row is 24 boys and 16 girls; when n=4, 12 and 8; n=8, 6 and 4; n=16, 3 and 2.Alternatively, when arranged in order from smallest to largest n, the per row numbers would be from highest to lowest. Either way, the problem doesn't specify the order, so as long as all possible divisors are listed.Thus, the final answer should be 4 different ways, with each row having (24 boys,16 girls), (12,8), (6,4), (3,2).Therefore, the answer is:Number of ways: 4Each arrangement's boys and girls per row: 3 and 2, 6 and 4, 12 and 8, 24 and 16.I need to check if there's another possible interpretation of the problem where "the number of boys and girls in each row being equal" might mean that each row has the same total number of children, combining boys and girls. But in that case, the problem would require that in each row, the total number is the same, but boys and girls could vary as long as the total is fixed. However, the problem specifically says "the number of boys and girls in each row being equal," which is a bit ambiguous.Wait, if the problem had meant that each row has the same total number of children, then we would have to find n such that 48 + 32 = 80 children are arranged into n rows, each with 80/n children. But that's a different problem, and the number of boys and girls per row could vary as long as the total per row is the same. But the problem states that "the number of boys and girls in each row being equal." So, perhaps that each row must have an equal number of boys and equal number of girls as in all rows have the same count of boys and the same count of girls, which is what I considered earlier.Alternatively, maybe each row has the same number of boys as girls. That is, in each row, boys = girls. So, if a row has k boys, it must have k girls. Then total boys would be k*n, total girls would be k*n. But since 48 ≠ 32, this is impossible. Therefore, this interpretation is invalid.Therefore, the correct interpretation must be that the number of boys per row is the same across all rows, and the number of girls per row is the same across all rows. Therefore, the problem reduces to finding the common divisors of 48 and 32 (excluding 1), which we found as 2,4,8,16. Thus, 4 arrangements.Hence, the answer is 4 ways, with each row's composition as follows:For n=2 rows: 24 boys and 16 girls per row.For n=4 rows: 12 boys and 8 girls per row.For n=8 rows: 6 boys and 4 girls per row.For n=16 rows: 3 boys and 2 girls per row.Therefore, the number of different arrangements is 4, and the corresponding boys and girls per row are the pairs listed above.Final AnswerThe number of different arrangements is boxed{4}, with each row containing the following numbers of boys and girls: boxed{3} boys and boxed{2} girls, boxed{6} boys and boxed{4} girls, boxed{12} boys and boxed{8} girls, boxed{24} boys and boxed{16} girls.</think>To solve the problem of arranging 48 boys and 32 girls into ( n ) rows (( n > 1 )) such that each row has the same number of boys and girls, we need to find the common divisors of 48 and 32 that are greater than 1. 1. Prime Factorization: - 48: ( 2^4 times 3 ) - 32: ( 2^5 )2. Greatest Common Divisor (GCD): - The GCD of 48 and 32 is ( 2^4 = 16 ).3. Divisors of GCD: - The divisors of 16 are 1, 2, 4, 8, and 16. Excluding 1, the valid divisors are 2, 4, 8, and 16.4. Number of Rows and Corresponding Boys/Girls per Row: - For each divisor ( n ): - ( n = 2 ): ( frac{48}{2} = 24 ) boys and ( frac{32}{2} = 16 ) girls per row. - ( n = 4 ): ( frac{48}{4} = 12 ) boys and ( frac{32}{4} = 8 ) girls per row. - ( n = 8 ): ( frac{48}{8} = 6 ) boys and ( frac{32}{8} = 4 ) girls per row. - ( n = 16 ): ( frac{48}{16} = 3 ) boys and ( frac{32}{16} = 2 ) girls per row.Thus, there are 4 different ways to arrange the students, with each row containing the specified number of boys and girls. Final AnswerThe number of different arrangements is boxed{4}, with each row containing the following numbers of boys and girls: boxed{3} boys and boxed{2} girls, boxed{6} boys and boxed{4} girls, boxed{12} boys and boxed{8} girls, boxed{24} boys and boxed{16} girls.,<think>Okay, let's try to solve this problem. So, we have 48 boys and 32 girls that need to be arranged in n rows, where n is more than 1. In each row, the number of boys and the number of girls must be equal. We need to find out how many different ways they can be arranged and specify how many boys and girls are in each row for each arrangement.First, let me understand the problem correctly. The ceremonial guard has 48 boys and 32 girls. They need to be arranged into n rows, and in each row, the number of boys and girls should be the same. So, each row must have the same number of boys and the same number of girls. But the total number of boys and girls in each row can vary between different arrangements, right? The key is that within each arrangement, every row has the same number of boys and girls.So, maybe we need to find the number of possible divisors that work for both boys and girls? Let me think. Since the number of boys and girls in each row has to be equal across all rows, the number of rows (n) must divide both the total number of boys and the total number of girls. Therefore, n has to be a common divisor of 48 and 32. But n has to be greater than 1, as specified.Wait, actually, the problem says they are arranged in n rows, with the number of boys and girls in each row being equal. So, perhaps the number of boys per row and the number of girls per row must each be the same across all rows. So, for each arrangement, we can have k boys per row and m girls per row, such that k is the same in every row and m is the same in every row, but k and m might be different. But the question says "the number of boys and girls in each row being equal". Hmm, maybe this is interpreted as the number of boys and girls per row being equal to each other? So, each row has the same number of boys as girls?Wait, the original problem says: "the number of boys and girls in each row being equal." So, perhaps in each row, the count of boys is equal to the count of girls. So, each row has the same number of boys and girls. For example, each row has 2 boys and 2 girls, or 3 boys and 3 girls, etc. Is that the correct interpretation?But if that's the case, then the total number of boys and girls must be the same, but here we have 48 boys and 32 girls. 48 is not equal to 32. Therefore, this interpretation might not be possible. So perhaps the problem is that in each row, the number of boys and the number of girls is the same across all rows, but not necessarily equal to each other. For example, each row has 6 boys and 4 girls, so that 6 boys times 8 rows would give 48 boys, and 4 girls times 8 rows would give 32 girls. So in each row, the number of boys is 6 and the number of girls is 4, and this is consistent across all rows.So, in this case, the number of boys per row and the number of girls per row must be divisors of 48 and 32 respectively, and the number of rows must be the same for both. So, the number of rows n must satisfy that n divides both 48 and 32. Therefore, n must be a common divisor of 48 and 32. Since n > 1, we need to find all common divisors of 48 and 32 greater than 1. The number of such divisors will give the number of different arrangements. For each such divisor n, the number of boys per row is 48/n and the number of girls per row is 32/n.Therefore, to solve the problem, we need to find the greatest common divisor (GCD) of 48 and 32, and then list all the divisors of that GCD, excluding 1. The number of such divisors will be the number of different ways they can be arranged, and each divisor corresponds to a number of rows. Then, for each divisor n, the number of boys per row is 48/n and the number of girls per row is 32/n.Let me verify this approach. Let's compute the GCD of 48 and 32. Prime factors of 48 are 2^4 * 3^1. Prime factors of 32 are 2^5. Therefore, the GCD is the minimum exponent for each prime, so 2^4 = 16. Therefore, the GCD of 48 and 32 is 16. Therefore, the common divisors of 48 and 32 are the divisors of 16. The divisors of 16 are 1, 2, 4, 8, 16. Since n > 1, the possible values for n are 2, 4, 8, 16. Therefore, there are 4 different ways to arrange them. For each n, the number of boys per row is 48/n and the number of girls per row is 32/n.Let's check for n=2: 48/2=24 boys per row, 32/2=16 girls per row. So each row has 24 boys and 16 girls. 24+16=40 people per row. With 2 rows, total boys:24*2=48, girls:16*2=32. That works.For n=4: 48/4=12 boys per row, 32/4=8 girls per row. Each row has 12 boys and 8 girls. 12+8=20 people per row. 4 rows: 12*4=48 boys, 8*4=32 girls. Correct.For n=8: 48/8=6 boys per row, 32/8=4 girls per row. Each row has 6 boys and 4 girls. 6+4=10 people per row. 8 rows: 6*8=48, 4*8=32. Correct.For n=16: 48/16=3 boys per row, 32/16=2 girls per row. Each row has 3 boys and 2 girls. 3+2=5 people per row. 16 rows: 3*16=48, 2*16=32. Correct.Therefore, the different arrangements are for n=2,4,8,16. So there are 4 different ways. For each arrangement, the number of boys and girls per row are as calculated above.Therefore, the answer should be 4 different ways, with the corresponding boys and girls per row as (24,16), (12,8), (6,4), (3,2).Wait, the question says "how many different ways can they be arranged? How many boys and girls are there in each row for each arrangement?" So we need to list the number of arrangements, which is 4, and for each arrangement, specify the number of boys and girls per row.But in the problem statement, the answer needs to be put into a box. However, the problem is in Chinese, but the instructions here just mention the problem. Wait, the original problem is in Chinese? Or is this the translation? Anyway, the user provided the problem in English, so the answer should be in English. The user mentions "put your final answer within boxed{}", but since the answer has two parts: the number of different ways and the number of boys and girls per row for each arrangement, maybe we need to present both.But in Chinese math problems, sometimes the answer is expected to list the number of ways and then list each possible arrangement. However, in the context of the user's question here, since they are asking for the number of different ways and the boys and girls per row for each arrangement, the answer would be:Number of ways: 4Arrangements:- 2 rows: 24 boys and 16 girls per row- 4 rows: 12 boys and 8 girls per row- 8 rows: 6 boys and 4 girls per row- 16 rows: 3 boys and 2 girls per rowBut the problem says "how many different ways can they be arranged? How many boys and girls are there in each row for each arrangement?" So both parts are required.But in the instructions, the user says "put your final answer within boxed{}". Usually, boxed{} is used for a single value. However, since there are two parts to the answer, maybe we need to present both in boxes. But perhaps in Chinese math problem conventions, it's acceptable to list multiple answers separated by commas or something. Alternatively, maybe the question expects the number of arrangements and then the possible per-row counts as pairs.But looking at the problem statement again: "How many different ways can they be arranged? How many boys and girls are there in each row for each arrangement?" So two questions. Therefore, the answer should have two parts: the number of ways, which is 4, and the details of each arrangement. Since the user wants the final answer in boxed notation, perhaps the first answer is 4, and the second is the list of pairs. But I need to check if there's a standard way to present such answers.Alternatively, maybe the problem expects the number of arrangements as the first boxed number and then the possible boys and girls per row as separate boxed numbers. But since there are four arrangements, each with different numbers, it's unclear. However, given that the original problem is worth 14 points, it's likely that both parts are required, and the answer should be structured accordingly.In Chinese math exams, when a question has two parts, you usually write both answers. Since the user mentioned "put your final answer within boxed{}", but the answer has two components, maybe we can box each part. For example:The number of different ways is boxed{4}, and the boys and girls in each row for each arrangement are boxed{3} and boxed{2}, boxed{6} and boxed{4}, boxed{12} and boxed{8}, boxed{24} and boxed{16}.But perhaps the order matters. Wait, in each case, the number of boys and girls per row are (3,2), (6,4), (12,8), (24,16) when arranged in increasing number of rows. However, the problem doesn't specify the order, so as long as all pairs are mentioned, it's okay. But in the answer, maybe list them in order from smallest number of rows to largest. Since n is 2,4,8,16, the per-row numbers go from 24,16 down to 3,2. So perhaps listing them from largest to smallest per-row counts.But I need to check if there's a specific way expected. Alternatively, maybe the answer is written as 4 ways, each corresponding to the divisors, and the pairs are (24,16), (12,8), (6,4), (3,2). Since the problem says "how many different ways can they be arranged?" which is 4, and "how many boys and girls are there in each row for each arrangement?" which are the four pairs.Therefore, to present the answer:The number of different ways is boxed{4}. For each arrangement, the number of boys and girls per row are boxed{24} and boxed{16}, boxed{12} and boxed{8}, boxed{6} and boxed{4}, boxed{3} and boxed{2}.But given that LaTeX boxes are usually for single answers, perhaps there is a different way. Alternatively, the answer might be written as boxed{4} ways with respective boys and girls per row: 24 & 16, 12 & 8, 6 & 4, 3 & 2. But the user instruction says "put your final answer within boxed{}", which is a bit ambiguous for multiple answers. However, given the context, I think the first answer is the number of ways, which is 4, and then each pair as separate boxed answers. But perhaps the problem expects all answers in one box. For example:boxed{4} arrangements: (24 boys, 16 girls), (12 boys, 8 girls), (6 boys, 4 girls), (3 boys, 2 girls) per row.But in Chinese math problems, when there are multiple answers, they might expect multiple boxes. However, since the platform here uses boxed{} for each answer, but the problem has two questions, maybe the user expects two boxed answers: the number of arrangements and the list of boys and girls per row. However, the list is multiple numbers. Alternatively, maybe the problem wants the number of ways and the possible number of boys and girls in each row, which would be the GCD's factors. Wait, but in the problem, the number of boys and girls per row is 48/n and 32/n, where n is a common divisor. Since the GCD is 16, the common divisors are 2,4,8,16, so the boys per row would be 24,12,6,3 and girls per row 16,8,4,2. So these are the pairs. Therefore, the answer is 4 ways, with those pairs. Therefore, the answer should be:Number of ways: boxed{4}For each arrangement, boys and girls per row: boxed{(24, 16)}, boxed{(12, 8)}, boxed{(6, 4)}, boxed{(3, 2)}But again, the user instruction says "put your final answer within boxed{}", which is confusing. Maybe the intended answer is simply the number of arrangements, which is 4, so boxed{4}, but that ignores the second part. Alternatively, maybe they expect all possible pairs listed in one box. However, the standard practice in math problems is to use separate boxes for separate answers. But given the user's instruction, maybe they want both answers in boxes. However, in the original problem statement, it's part 13 worth 14 points, so it's a substantial question. Therefore, it's more likely that the answer should be written as two parts: the number of ways and the list of boys and girls per row. But given the requirement here to put the final answer within boxed{}, maybe the answer is written as two boxed parts. For example:The number of arrangements is boxed{4}, and the possible boys and girls per row are boxed{3}, boxed{2}; boxed{6}, boxed{4}; boxed{12}, boxed{8}; boxed{24}, boxed{16}.But this seems messy. Alternatively, if there's a standard way to represent multiple answers, perhaps separated by commas. But I need to check if the platform supports multiple boxes. On platforms like Brilliant or in certain problem formats, sometimes multiple boxes are used. For example, if the answer is a list, each element can be boxed. Alternatively, if it's two separate answers, they can be boxed separately.But given the problem asks two questions: "How many different ways can they be arranged? How many boys and girls are there in each row for each arrangement?" So two questions, so two answers. Therefore, the first answer is 4, and the second answer is the list of pairs. However, since the user instruction says "put your final answer within boxed{}", perhaps both answers are expected in separate boxes. But the user might be expecting the first answer as a single boxed number, and the second as a list of boxed pairs.Alternatively, since in Chinese math problems, sometimes answers are written in a specific order with boxes. Maybe the answer should be written as:The number of different arrangements is boxed{4}, with each row containing boxed{3} boys and boxed{2} girls, boxed{6} boys and boxed{4} girls, boxed{12} boys and boxed{8} girls, or boxed{24} boys and boxed{16} girls.But again, this is a bit unconventional. Alternatively, maybe just present the number of ways as boxed{4}, and for the second part, list all the possible pairs. However, since the user specified "put your final answer within boxed{}", perhaps they expect all answers to be inside a single box, separated appropriately. But I don't recall seeing such usage. Alternatively, given that the problem is worth 14 points, and likely has 14 points allocated for showing work and correct answers, but in this context, the assistant is to provide the final answer boxed.Given the ambiguity, but considering standard practice, the number of arrangements is 4, which can be boxed, and the pairs can be presented as boxed pairs. However, since the user hasn't specified how to handle multiple answers, but in previous examples, when there are multiple answers, they are each boxed. For example, if the answer is a list, each element is boxed. Alternatively, if there are two answers, they are both boxed. Since the problem has two questions, the answer is two parts. Therefore, the first answer is 4, and the second answer is the four pairs. However, given the limitation of the platform, perhaps we need to box each pair as separate answers. But given that the user hasn't specified, perhaps the safest way is to box the number of arrangements and mention the pairs in words. But the user might expect the pairs to be boxed as well. Alternatively, since the problem might be from a math competition, where answers are often numerical, but here the second part is a set of pairs, which can't be boxed numerically. Hmm.Alternatively, since the problem might be in Chinese, maybe the answer expects the number of arrangements and then the possible number of people per row. Wait, no. The question explicitly asks for the number of boys and girls in each row for each arrangement.Wait, perhaps the answer is 4 different ways, and for each way, the number of boys and girls per row is determined by the common divisors. Therefore, the answer is 4, and the pairs are (24,16), (12,8), (6,4), (3,2). In the absence of specific instructions, and given that the user wants the answer within boxed{}, perhaps we can write the number of ways as boxed{4} and then list the pairs in parentheses. But since the platform might not accept multiple boxes or non-numerical answers in boxes, maybe the answer is simply boxed{4} for the number of ways, and the pairs are described in words. However, the original problem requires both answers.Alternatively, perhaps the answer is presented as boxed{4} arrangements with the corresponding boys and girls per row as boxed{(3,2)}, boxed{(6,4)}, boxed{(12,8)}, and boxed{(24,16)}. This way, each pair is boxed. But this seems a bit excessive, but maybe acceptable.Given the uncertainty, but wanting to comply with the user's instruction, I will present the number of arrangements as a boxed number and each pair of boys and girls per row as boxed pairs separated by commas.Final AnswerThe number of different ways is boxed{4}. The boys and girls in each row for each arrangement are boxed{24} and boxed{16}, boxed{12} and boxed{8}, boxed{6} and boxed{4}, boxed{3} and boxed{2}.</think>To solve the problem of arranging 48 boys and 32 girls into ( n ) rows (( n > 1 )) with equal numbers of boys and girls in each row, we need to find the common divisors of 48 and 32, excluding 1. First, we determine the greatest common divisor (GCD) of 48 and 32. The prime factorization of 48 is ( 2^4 times 3 ) and the prime factorization of 32 is ( 2^5 ). The GCD is ( 2^4 = 16 ).The common divisors of 48 and 32 are the divisors of 16, which are 1, 2, 4, 8, and 16. Excluding 1, the valid values for ( n ) are 2, 4, 8, and 16. For each valid ( n ):- When ( n = 2 ), each row has ( frac{48}{2} = 24 ) boys and ( frac{32}{2} = 16 ) girls.- When ( n = 4 ), each row has ( frac{48}{4} = 12 ) boys and ( frac{32}{4} = 8 ) girls.- When ( n = 8 ), each row has ( frac{48}{8} = 6 ) boys and ( frac{32}{8} = 4 ) girls.- When ( n = 16 ), each row has ( frac{48}{16} = 3 ) boys and ( frac{32}{16} = 2 ) girls.Thus, there are 4 different ways to arrange them, and the number of boys and girls in each row for each arrangement are:The number of different ways is (boxed{4}). The boys and girls in each row for each arrangement are (boxed{24}) and (boxed{16}), (boxed{12}) and (boxed{8}), (boxed{6}) and (boxed{4}), (boxed{3}) and (boxed{2}).]

🔑:13. (14 points) To welcome the "World Women's Conference", a school has formed a ceremonial team consisting of 48 boys and 32 girls. They need to be arranged into n rows (n>1), with the number of boys and girls in each row being equal. How many different arrangements are possible? How many boys and girls are there in each row for each arrangement?【Solution】Solution: The common divisors of 48 and 32 are 2, 4, 8, 16. Therefore, there are 4 possible arrangements:(1) Arrange in 2 rows: Each row has 24 boys and 16 girls;(2) Arrange in 4 rows: Each row has 12 boys and 8 girls;(3) Arrange in 8 rows: Each row has 6 boys and 4 girls;(4) Arrange in 16 rows: Each row has 3 boys and 2 girls;Answer: There are 4 possible arrangements. When arranged in 2 rows, each row has 24 boys and 16 girls; when arranged in 4 rows, each row has 12 boys and 8 girls; when arranged in 8 rows, each row has 6 boys and 4 girls; when arranged in 16 rows, each row has 3 boys and 2 girls.

🧠:[<think>Okay, so I need to find a fraction that has a denominator of 20, and its value has to be between -5/13 and -4/13. Hmm, let me think about how to approach this. First, I remember that fractions can be converted to decimals to make comparisons easier. Let me start by converting -5/13 and -4/13 to decimal form. That might help me figure out what range the new fraction should be in.Calculating -5/13: So, 5 divided by 13. Let me do that division. 13 goes into 5 zero times, so we add a decimal point and a zero, making it 50. 13 goes into 50 three times (13*3=39), subtract 39 from 50, you get 11. Bring down another zero to make 110. 13 goes into 110 eight times (13*8=104), subtract 104 from 110, get 6. Bring down another zero to make 60. 13 goes into 60 four times (13*4=52), subtract 52 from 60, get 8. Bring down another zero to make 80. 13 goes into 80 six times (13*6=78), subtract 78 from 80, get 2. Bring down another zero to make 20. 13 goes into 20 once (13*1=13), subtract 13 from 20, get 7. Bring down another zero to make 70. 13 goes into 70 five times (13*5=65), subtract 65 from 70, get 5. Hmm, this is starting to repeat. So, 5 divided by 13 is approximately 0.384615... So, -5/13 is approximately -0.384615...Similarly, let's do -4/13. 4 divided by 13. 13 into 40 three times (13*3=39), remainder 1. Bring down a zero to make 10. 13 into 10 zero times, so we have 0.3... Then bring down another zero to make 100. 13 into 100 seven times (13*7=91), subtract 91 from 100, get 9. Bring down another zero to make 90. 13 into 90 six times (13*6=78), subtract 78 from 90, get 12. Bring down another zero to make 120. 13 into 120 nine times (13*9=117), subtract 117, get 3. Bring down another zero to make 30. 13 into 30 twice (13*2=26), subtract 26 from 30, get 4. Bring down another zero to make 40. Now we're back to where we started. So, 4 divided by 13 is approximately 0.307692... So, -4/13 is approximately -0.307692...So the value we need is between approximately -0.384615... and -0.307692...Now, the fraction we're looking for must have a denominator of 20. Let's denote this fraction as -n/20, where n is a positive integer (since the original fractions are negative, the desired fraction is also negative). So we need to find an integer n such that:-5/13 < -n/20 < -4/13But when we deal with inequalities involving negatives, we have to remember that multiplying or dividing by a negative number reverses the inequality sign. However, in this case, since all terms are negative, maybe it's easier to reverse the inequalities by taking the reciprocal and considering positive numbers. Wait, no. Let me just work directly with the negatives.Alternatively, if we remove the negatives by multiplying all parts by -1, but this reverses the inequalities:5/13 > n/20 > 4/13So that becomes:4/13 < n/20 < 5/13So n/20 must be between 4/13 and 5/13. Then, multiplying all parts by 20 to solve for n:(4/13)*20 < n < (5/13)*20Calculating (4/13)*20: 4*20=80, 80/13 ≈ 6.1538...Calculating (5/13)*20: 5*20=100, 100/13 ≈ 7.6923...So n must be greater than approximately 6.1538 and less than approximately 7.6923. Since n has to be an integer (because the numerator must be an integer for the fraction to have denominator 20), the possible integers n can be are 7.Wait, 6.1538 < n < 7.6923. But n must be an integer. So the integers between 6.1538 and 7.6923 are 7. So n=7. Therefore, the fraction is -7/20.But wait, let me check that. Let me confirm if -7/20 is indeed between -5/13 and -4/13.First, compute -5/13 ≈ -0.3846, -7/20 = -0.35, and -4/13 ≈ -0.3077. Wait, but -0.35 is between -0.3846 and -0.3077? Let me see:-0.3846 (which is -5/13) < -0.35 < -0.3077 (which is -4/13)? Wait, no. Because -0.35 is actually greater than -0.3846 but less than -0.3077? Wait, in negatives, the number with the smaller absolute value is larger. So, since 0.35 is between 0.3077 and 0.3846? Wait, no. Wait, 0.35 is between 0.3077 and 0.3846? Let's check:0.3077 (which is 4/13) ≈ 0.3077, 0.35, 0.3846 (which is 5/13). So 0.35 is between 0.3077 and 0.3846, which would mean that -0.35 is between -0.3846 and -0.3077. But in the negative direction, the order is reversed. Let me think: if a and b are positive numbers with a < b, then -a > -b. So, since 0.3077 < 0.35 < 0.3846, then -0.3077 > -0.35 > -0.3846. So, yes, -0.35 is indeed between -0.3846 and -0.3077. Therefore, -7/20 is between -5/13 and -4/13. Therefore, -7/20 is the answer.But wait, let me check if there are other possible fractions. Because in the calculation above, n must be between approximately 6.1538 and 7.6923. So the integer values are 7. But what if there are other fractions? Wait, denominator is 20, so the numerator has to be an integer, so possible numerators are integers from 1 to 19 (since the fraction is negative). But according to the calculation, n must be approximately between 6.15 and 7.69, so only 7. So only -7/20 is the possible fraction. But let me confirm this.Alternatively, maybe cross-multiplying the inequalities to avoid decimal approximations.Starting with -5/13 < -n/20 < -4/13.Multiply all parts by -1, reversing inequalities:5/13 > n/20 > 4/13So 4/13 < n/20 < 5/13Multiply all parts by 20:(4/13)*20 < n < (5/13)*20Calculating (4*20)/13 = 80/13 ≈6.1538...(5*20)/13=100/13≈7.6923...So n must be between approximately 6.1538 and 7.6923. Since n must be an integer, only 7 is in that interval. Therefore, n=7. Hence, the fraction is -7/20.But to confirm, let's check -7/20.Convert -7/20 to a decimal: 7 divided by 20 is 0.35, so -0.35.Earlier, we found that -5/13 ≈ -0.3846 and -4/13≈-0.3077. So, -0.3846 < -0.35 < -0.3077? Wait, let's check the actual values.-0.3846 is approximately -5/13. Then, -0.35 is in between? Wait, no. Wait, in terms of magnitude:0.3846 is bigger than 0.35, so -0.3846 is less than -0.35. Then, -0.35 is greater than -0.3846 but less than -0.3077?Wait, let me think again. On the number line, going from left to right: -0.3846, -0.35, -0.3077. So -0.35 is to the right of -0.3846 and to the left of -0.3077. Therefore, yes, -0.35 is between -5/13 and -4/13. Therefore, -7/20 is between them.But just to make sure, let's check using fractions instead of decimals.Compare -5/13 and -7/20. To see which is bigger, cross-multiply.-5/13 vs. -7/20.Multiply both sides by 13*20 to eliminate denominators:-5*20 vs. -7*13-100 vs. -91Since -100 < -91, this means that -5/13 < -7/20.Similarly, compare -7/20 and -4/13.Again cross-multiplying:-7/20 vs. -4/13Multiply both sides by 20*13:-7*13 vs. -4*20-91 vs. -80Since -91 < -80, this means that -7/20 < -4/13. Wait, that contradicts the earlier decimal approach. Wait, so according to this cross-multiplication, -7/20 is less than -4/13? But according to the decimal, -7/20 is -0.35, and -4/13≈-0.3077, so -0.35 is less than -0.3077. That would mean that -7/20 is actually less than -4/13, which is not between them. Wait, now I'm confused.Wait, that can't be. Wait, if cross-multiplying gives -7/20 < -4/13, that would mean that -7/20 is further to the left on the number line, i.e., smaller. So, if -5/13 ≈-0.3846, -7/20=-0.35, and -4/13≈-0.3077, then:-0.3846 (which is -5/13) is the smallest, then -0.35 (which is -7/20), then -0.3077 (which is -4/13). So, from smallest to largest: -5/13, -7/20, -4/13. Therefore, -7/20 is between them. But when cross-multiplying, comparing -7/20 and -4/13, we get:Cross-multiplying: -7*13 = -91 and -4*20 = -80. Since -91 < -80, then -7/20 < -4/13. Therefore, this suggests that -7/20 is less than -4/13, which would mean it's not between them. But according to the decimal, it is between them. So which one is correct?Wait, here's the problem. When cross-multiplying inequalities with negative numbers, we have to be careful with the sign.Let me clarify: when comparing two fractions, a/b and c/d, if a/b and c/d are both positive, then a/b > c/d if and only if ad > bc. However, if both are negative, the inequality reverses. Wait, actually, let's think carefully.Suppose we have two negative fractions: -a/b and -c/d, where a, b, c, d are positive.Which one is larger? The one with the smaller absolute value. So, -a/b > -c/d if and only if a/b < c/d.Therefore, when comparing -7/20 and -4/13, we can compare their absolute values: 7/20 and 4/13.Compute 7/20 = 0.35, 4/13 ≈0.3077. Since 0.35 > 0.3077, then 7/20 > 4/13, therefore, -7/20 < -4/13. So that's why cross-multiplying gives that -7/20 is less than -4/13. But according to the decimal equivalents, -7/20 is -0.35, which is between -0.3846 and -0.3077. Wait, no. If -7/20 is -0.35, which is between -0.3846 and -0.3077, that would mean that -0.3846 < -0.35 < -0.3077. But in reality, -0.35 is less than -0.3077, so -0.35 is not between them. Wait, no! Wait, this is conflicting.Wait, maybe my confusion arises from mixing up the decimal values.Wait, let's list the numbers in order from least to greatest.Original numbers: -5/13 ≈ -0.3846, -4/13 ≈ -0.3077.But where does -0.35 fit? -0.35 is equivalent to -7/20. So, since -0.3846 is less than -0.35, which is less than -0.3077? Wait, no. Negative numbers increase towards zero. So, -0.3846 is less than -0.35, which is less than -0.3077. Wait, but -0.35 is actually between -0.3846 and -0.3077.Wait, here's the correct order:Numbers on the number line:-0.3846 (-5/13) ---- -0.35 (-7/20) ---- -0.3077 (-4/13) ---- 0So, moving from left to right (from smaller to larger):-5/13 < -7/20 < -4/13.But according to cross-multiplication:Comparing -7/20 and -4/13, cross-multiplying gives -91 < -80, so -7/20 < -4/13.But on the number line, -7/20 is to the right of -5/13 and to the left of -4/13. So, in terms of order, -5/13 < -7/20 < -4/13. But cross-multiplication says -7/20 < -4/13. Wait, that can't be both. Wait, but according to the decimal, -0.35 is greater than -0.3846, but less than -0.3077. Wait, no. -0.35 is less than -0.3077. Because -0.35 is more negative than -0.3077. Wait, no. Wait, if you have two numbers, -0.35 and -0.3077, which is larger? Since -0.3077 is closer to zero, it's larger. So, -0.35 is less than -0.3077. Therefore, -7/20 is less than -4/13. Therefore, the correct order is -5/13 < -7/20 < -4/13. Wait, that can't be. Because if -7/20 is less than -4/13, then it's further left on the number line, so between -5/13 and -4/13, but less than both. Wait, but -5/13 is already less than -7/20. Wait, no, this is confusing.Wait, let's get back to basics. Let's convert all fractions to decimals:-5/13 ≈ -0.3846-7/20 = -0.35-4/13 ≈ -0.3077So, in decimal form:-0.3846, -0.35, -0.3077So, on the number line:----- -0.3846 ----- -0.35 ----- -0.3077 ----- 0So, from left to right: -5/13, -7/20, -4/13.Therefore, -5/13 < -7/20 < -4/13 is false. Because in reality, -7/20 is greater than -5/13 and less than -4/13. Wait, but decimals say otherwise. Wait, no. Wait, the confusion comes from the fact that -0.35 is actually greater than -0.3846, because it's closer to zero. Similarly, -0.35 is less than -0.3077, because it's further from zero. So, actually, the order is:-0.3846 < -0.35 < -0.3077But this is not correct because -0.35 is greater than -0.3846 but less than -0.3077. Wait, how?Wait, numbers increase from left to right on the number line. So:-0.3846 (which is -5/13) is to the left of -0.35 (-7/20), which is to the left of -0.3077 (-4/13). Therefore:-5/13 < -7/20 < -4/13But since -7/20 is between them, it should be the case that -5/13 < -7/20 AND -7/20 < -4/13. But according to cross-multiplication:First inequality: -5/13 < -7/20Multiply both sides by 13*20 (which is positive, so inequality remains the same):-5*20 < -7*13-100 < -91Which is true, because -100 is to the left of -91.Second inequality: -7/20 < -4/13Multiply both sides by 13*20:-7*13 < -4*20-91 < -80Which is also true, since -91 is to the left of -80.Therefore, both inequalities hold. Therefore, -5/13 < -7/20 < -4/13 is indeed true.Therefore, even though -7/20 is less than -4/13, it's still between -5/13 and -4/13 because -5/13 is less than -7/20, which is less than -4/13. Wait, no. Wait, if -5/13 < -7/20 and -7/20 < -4/13, then combining these, we have -5/13 < -7/20 < -4/13, which would mean that -7/20 is between -5/13 and -4/13. Therefore, the fraction -7/20 is indeed between the two given fractions.Therefore, the answer is -7/20.But let me verify once more.Original problem: find a fraction with denominator 20 between -5/13 and -4/13.We found -7/20. Let's check:First, compare -5/13 and -7/20.Convert to decimals:-5/13 ≈ -0.3846-7/20 = -0.35So, -0.3846 < -0.35, which is true. Then compare -0.35 and -4/13≈-0.3077. Since -0.35 < -0.3077, this is also true. Therefore, -0.3846 < -0.35 < -0.3077, so -5/13 < -7/20 < -4/13. Therefore, -7/20 is between them. Hence, correct.Therefore, the answer is -7/20.But wait, just to ensure that there isn't another fraction with denominator 20 between them. Let's see. The next possible numerators would be 6 and 8. Let's check:-6/20 simplifies to -3/10 = -0.3. Is -0.3 between -0.3846 and -0.3077? -0.3 is greater than -0.3077, so no. So -6/20 is -0.3, which is outside the upper bound.-8/20 simplifies to -2/5 = -0.4. Comparing to -5/13≈-0.3846. -0.4 is less than -0.3846, so it's outside the lower bound. Therefore, -8/20 is too small.Thus, only -7/20 is between them. Therefore, the answer is boxed{-dfrac{7}{20}}.Final Answerboxed{-dfrac{7}{20}}</think>To find a fraction with a denominator of 20 that is between (-frac{5}{13}) and (-frac{4}{13}), we start by converting these fractions to their decimal forms:- (-frac{5}{13} approx -0.3846)- (-frac{4}{13} approx -0.3077)We need a fraction (-frac{n}{20}) such that (-0.3846 < -frac{n}{20} < -0.3077). Converting this to positive values, we get:- (frac{4}{13} < frac{n}{20} < frac{5}{13})Multiplying all parts by 20 to solve for (n):- (frac{4}{13} times 20 < n < frac{5}{13} times 20)- (frac{80}{13} approx 6.1538 < n < frac{100}{13} approx 7.6923)The only integer (n) in this range is 7. Therefore, the fraction is (-frac{7}{20}).To verify, we convert (-frac{7}{20}) to a decimal:- (-frac{7}{20} = -0.35)Checking the order on the number line:- (-0.3846 < -0.35 < -0.3077), which confirms that (-frac{7}{20}) is indeed between (-frac{5}{13}) and (-frac{4}{13}).Thus, the final answer is (boxed{-dfrac{7}{20}}).,<think>Okay, so I need to find a fraction with a denominator of 20 that lies between -5/13 and -4/13. Hmm, let me think. First, I remember that when comparing fractions, it's helpful to convert them to decimals or have a common denominator. Since the target fraction needs a denominator of 20, maybe converting the given fractions to have a denominator of 20 could help. Wait, but 13 and 20 don't have common factors, so maybe converting to decimals is easier?Let me calculate -5/13 as a decimal. Let's see, 5 divided by 13. 13 goes into 5 zero times, add a decimal point, 13 into 50 is 3 times (since 13*3=39), subtract 39 from 50, we get 11. Bring down the next 0, making it 110. 13 into 110 is 8 times (13*8=104), subtract 104 from 110, remainder 6. Bring down a 0, making it 60. 13 into 60 is 4 times (13*4=52), subtract 52 from 60, remainder 8. Bring down another 0, making it 80. 13 into 80 is 6 times (13*6=78), remainder 2. Bring down a 0, making it 20. 13 into 20 is 1 time (13*1=13), remainder 7. Bring down a 0, making it 70. 13 into 70 is 5 times (13*5=65), remainder 5. Wait, this is starting to repeat. So, 5/13 is approximately 0.384615..., so -5/13 is approximately -0.384615.Similarly, let's compute -4/13. 4 divided by 13. 13 into 40 is 3 times (13*3=39), remainder 1. Bring down a 0, making it 10. 13 into 10 is 0 times, so add a decimal and a zero. 13 into 100 is 7 times (13*7=91), remainder 9. Bring down a 0, making it 90. 13 into 90 is 6 times (13*6=78), remainder 12. Bring down a 0, making it 120. 13 into 120 is 9 times (13*9=117), remainder 3. Bring down a 0, making it 30. 13 into 30 is 2 times (13*2=26), remainder 4. Bring down a 0, making it 40. Now we see the cycle starts again. So, 4/13 ≈ 0.307692..., so -4/13 is approximately -0.307692.So we need a fraction with denominator 20 between approximately -0.384615 and -0.307692. Let's list the fractions with denominator 20 in that range. Remember that fractions with denominator 20 are multiples of 1/20. Let's convert 1/20 to decimal: 0.05. So the fractions are -0.05, -0.10, -0.15, -0.20, -0.25, -0.30, -0.35, -0.40, etc.But wait, we need to find a negative fraction between -5/13 (-0.3846) and -4/13 (-0.3077). So, in decimal terms, the value needs to be between approximately -0.3846 and -0.3077. Let's check the negative fractions with denominator 20:-1/20 = -0.05 (too small, not negative enough)-2/20 = -0.10 (still too small)-3/20 = -0.15 (still too small)-4/20 = -0.20 (too small)-5/20 = -0.25 (still too small)-6/20 = -0.30 (hmm, -0.30 is -0.30, which is approximately -3/10. How does that compare to -4/13 ≈ -0.3077? Since -0.30 is greater than -0.3077, so -6/20 is actually greater than -4/13, which is not between -5/13 and -4/13. Wait, we need a number that is less than -4/13 but greater than -5/13. So between the two negatives, which one is bigger? Since -0.3846 is less than -0.3077, so the number must be between -0.3846 and -0.3077. So numbers like -0.31, -0.32, etc. But in terms of denominator 20, the fractions are in increments of 0.05. Let's check:-7/20 = -0.35. Is -0.35 between -0.3846 and -0.3077? Let's see. -0.3846 < -0.35 < -0.3077? Let's check. Since -0.35 is greater than -0.3846 (because it's less negative) and -0.35 is less than -0.3077 (because it's more negative)? Wait, no. Wait, in negative numbers, the number with larger absolute value is smaller. So, -0.35 is more negative than -0.3077, so -0.35 is less than -0.3077. So -7/20 is -0.35 which is less than -0.3077, so that's outside the upper bound. Then, -6/20 is -0.30, which is greater than -0.3077, so it's above the upper bound. Hmm, so -6/20 is -0.30, which is actually greater than -4/13 (-0.3077). So -0.30 is greater than -0.3077? Let me verify. Yes, because 0.30 is less than 0.3077, so when negated, -0.30 is greater than -0.3077. So -6/20 is -0.30, which is greater than -4/13. Wait, but we need a number between -5/13 (-0.3846) and -4/13 (-0.3077). So, numbers like -0.31, -0.32, up to -0.38. But with denominator 20, the possible numerators are integers. So possible fractions are multiples of 0.05. The fractions between -0.3846 and -0.3077 would need to be between approximately -0.38 and -0.31. But the closest 20 denominator fractions would be -7/20 (-0.35), -8/20 (-0.40), but wait. Wait, -0.35 is -7/20, which is -0.35. Is -0.35 between -0.3846 and -0.3077? Let's check: -0.3846 < -0.35 < -0.3077? Since -0.35 is greater than -0.3846 (because it's to the right on the number line) and -0.35 is less than -0.3077 (to the left). Wait, but -0.35 is less than -0.3077? No. Wait, in decimal terms, -0.35 is -0.3500, which is less than -0.3077, right? Because -0.35 is more negative. So, no. -0.35 is not between -0.3846 and -0.3077. Then, what about -8/20? That's -0.40, which is even smaller. That's outside the lower bound. So maybe there's no such fraction? But that can't be. The problem says to find one. Maybe I made a mistake.Wait, perhaps converting to decimals isn't the best approach. Maybe I should convert both fractions to have a common denominator with 20, but since 13 and 20 are coprime, the common denominator would be 260. Let's try that.So, the two fractions are -5/13 and -4/13. Let's convert them to denominator 260. 13*20=260. So, -5/13 = (-5*20)/260 = -100/260. Similarly, -4/13 = (-4*20)/260 = -80/260. The target fraction is x/20, which is equivalent to (x*13)/260. So, we need (x*13)/260 between -100/260 and -80/260. Therefore, x*13 must be between -100 and -80. So, -100 < 13x < -80. Solving for x: divide all parts by 13. So, -100/13 ≈ -7.6923 < x < -80/13 ≈ -6.1538. Since x must be an integer (because the numerator of the fraction with denominator 20 must be an integer), the integer x must satisfy -7.6923 < x < -6.1538. So the integers in that interval are x = -7. Because -7 is greater than -7.6923 and less than -6.1538. Wait, wait. Let's check: -7 is approximately -7.0. So -7.6923 < -7.0 < -6.1538. Yes, because -7.0 is between -7.6923 and -6.1538. Therefore, x = -7. So the fraction is -7/20. Let's check if that's correct.Convert -7/20 to decimal: -0.35. Then, check if -0.35 is between -5/13 ≈ -0.3846 and -4/13 ≈ -0.3077. Wait, -0.35 is greater than -0.3846 (since it's closer to zero) and less than -0.3077? Wait, no. -0.35 is actually less than -0.3077, because it's more negative. So -0.35 is not between -0.3846 and -0.3077. Wait, that's conflicting with the previous calculation. What's going on here?Hold on, there seems to be a discrepancy. When converting to denominator 260, we found x must be -7, so -7/20. But converting that to decimal is -0.35, which is outside the upper bound. So that's a problem. Where did I go wrong?Let me re-examine the steps. Converting -5/13 and -4/13 to denominator 260 gives -100/260 and -80/260. The target fraction is x/20 = (13x)/260. We need (13x) to be between -100 and -80. So, -100 < 13x < -80. Dividing by 13: -100/13 ≈ -7.6923 < x < -80/13 ≈ -6.1538. So x must be an integer between approximately -7.6923 and -6.1538. The integers in that range are x = -7. Because -7 is -7.0, which is greater than -7.6923 and less than -6.1538. So, x = -7. Therefore, the fraction is -7/20. But -7/20 is -0.35, which is less than -4/13 ≈ -0.3077, so it's actually outside the upper bound. That's conflicting. So why did the algebra say it's between? Maybe there's a miscalculation here.Wait, let's check the inequalities again. We have:Original fractions: -5/13 ≈ -0.3846 and -4/13 ≈ -0.3077.We need x/20 between them. So, -5/13 < x/20 < -4/13.Multiplying all parts by 20: (-5/13)*20 < x < (-4/13)*20.Calculating (-5/13)*20: (-100)/13 ≈ -7.6923.Calculating (-4/13)*20: (-80)/13 ≈ -6.1538.Therefore, x must be between approximately -7.6923 and -6.1538. Since x must be an integer, the only integer in that interval is x = -7. Therefore, x = -7 gives the fraction -7/20. But when we convert that to decimal, it's -0.35, which is less than -4/13 (≈-0.3077). Wait, that's not between them. So, perhaps there is a mistake here.Wait, but -7.6923 < x < -6.1538. So x is a number between approximately -7.69 and -6.15. Since x has to be an integer, the only integer is -7. So x = -7. Then, x/20 = -7/20 = -0.35. But -0.35 is not between -0.3846 and -0.3077. Wait, -0.35 is greater than -0.3846 (because -0.35 is to the right of -0.3846 on the number line), but -0.35 is less than -0.3077 (because it's to the left). So, -0.35 is not between them. Hmm. So, perhaps there is no such fraction? But the problem states to find one. So maybe my approach is wrong.Alternatively, maybe I made a mistake in setting up the inequality. Let's re-examine. The problem says the fraction with denominator 20 is between -5/13 and -4/13. So, we need:-5/13 < x/20 < -4/13.Multiply all parts by 20:-5/13 * 20 < x < -4/13 * 20.Calculating:-5*20/13 = -100/13 ≈ -7.6923.-4*20/13 = -80/13 ≈ -6.1538.Therefore, x must be a number between approximately -7.6923 and -6.1538. Since x must be an integer (as numerator of a fraction with denominator 20), the only integer in that interval is -7. So x = -7. Then, x/20 = -7/20 = -0.35. But -0.35 is not between -5/13 ≈ -0.3846 and -4/13 ≈ -0.3077. Wait, so this suggests that there is no such fraction? But the problem says to find one, so maybe there's a miscalculation here. Alternatively, perhaps the question allows for non-integer numerators? But no, fractions have integer numerators. So maybe the problem is designed to have such a fraction even if in decimal terms it seems outside? Wait, perhaps when comparing fractions, cross-multiplying to check.Let's check if -5/13 < -7/20 < -4/13.First, check -5/13 < -7/20.Multiply both sides by 13*20 (which is positive, so inequality direction remains):-5*20 < -7*13-100 < -91Which is true, because -100 is less than -91.Second, check -7/20 < -4/13.Multiply both sides by 13*20:-7*13 < -4*20-91 < -80Which is also true, since -91 is less than -80.Wait, so according to cross-multiplication, -7/20 is indeed between -5/13 and -4/13. But in decimal terms, we have -0.3846 < -0.35 < -0.3077, which contradicts, because -0.35 is not between them. But cross-multiplication says it is. So there's a contradiction here? Wait, no. Wait, hold on. Let's get this straight.Wait, in decimal terms:-5/13 ≈ -0.3846-7/20 = -0.35-4/13 ≈ -0.3077So, on the number line:-0.3846 (left) ---- -0.35 ---- -0.3077 (right)Wait, but -0.35 is to the right of -0.3846 and to the left of -0.3077. Wait, no. On the number line, numbers increase to the right. So:<---- (-0.3846) ---- (-0.35) ---- (-0.3077) ---- 0 ---->So, -0.35 is between -0.3846 and -0.3077. Because -0.3846 is the smallest (most negative), then -0.35, then -0.3077. So yes, -0.35 is between them. Wait, but in terms of magnitude, -0.35 is greater than -0.3846 (since it's less negative) and less than -0.3077 (since it's more negative than -0.3077). Wait, no. Wait, if you have two negative numbers, the one closer to zero is larger. So:-0.3846 < -0.35 < -0.3077.Wait, no. -0.35 is actually greater than -0.3846, because it's closer to zero, and less than -0.3077? Wait, no. Wait, let me clarify:Numbers on the number line:-0.3846, -0.35, -0.3077.Order from left to right (smallest to largest):-0.3846, -0.35, -0.3077.Therefore, -0.35 is greater than -0.3846 and less than -0.3077. Wait, no. How does that work? If you have three numbers: a = -0.3846, b = -0.35, c = -0.3077.Since they are negative numbers, a < b < c because as you move to the right on the number line, the numbers increase. So, a (-0.3846) is less than b (-0.35), which is less than c (-0.3077). Therefore, -0.35 is between -0.3846 and -0.3077. Wait, that's the opposite of what I thought earlier. So maybe my confusion was due to thinking about the magnitudes. But in terms of actual value, -0.35 is indeed between -0.3846 and -0.3077. Because:-0.3846 < -0.35 < -0.3077.Yes, that's correct. Because -0.35 is to the right of -0.3846 and to the left of -0.3077. So it is in between. So even though in absolute terms, 0.35 is between 0.3077 and 0.3846, when negated, their order reverses. So, -0.35 is between -0.3846 and -0.3077. Therefore, -7/20 is indeed between -5/13 and -4/13. Therefore, the answer is -7/20. So my initial confusion was from misapplying the decimal comparisons, but cross-multiplication confirms it. So the fraction is -7/20.Let me verify once more using cross-multiplication:First, check if -5/13 < -7/20:Multiply -5 * 20 = -100Multiply -7 * 13 = -91Since -100 < -91, the inequality -5/13 < -7/20 holds true.Second, check if -7/20 < -4/13:Multiply -7 * 13 = -91Multiply -4 * 20 = -80Since -91 < -80, the inequality -7/20 < -4/13 also holds true.Therefore, -7/20 is indeed between -5/13 and -4/13.So the answer is boxed{-dfrac{7}{20}}.Final Answerboxed{-dfrac{7}{20}}</think>To find a fraction with a denominator of 20 between (-frac{5}{13}) and (-frac{4}{13}), we start by converting the given fractions to decimals for easier comparison:- (-frac{5}{13} approx -0.3846)- (-frac{4}{13} approx -0.3077)We need a fraction (-frac{x}{20}) (where (x) is a positive integer) such that:[-0.3846 < -frac{x}{20} < -0.3077]Multiplying all parts by 20 to clear the denominator:[-0.3846 times 20 < -x < -0.3077 times 20]Calculating the products:[-7.6923 < -x < -6.1538]Multiplying through by -1 (reversing the inequalities):[6.1538 < x < 7.6923]The only integer (x) in this range is 7. Therefore, the fraction is:[-frac{7}{20}]Verifying using cross-multiplication:- Check (-frac{5}{13} < -frac{7}{20}): [ -5 times 20 = -100 quad text{and} quad -7 times 13 = -91 quad text{since} quad -100 < -91 ]- Check (-frac{7}{20} < -frac{4}{13}): [ -7 times 13 = -91 quad text{and} quad -4 times 20 = -80 quad text{since} quad -91 < -80 ]Both inequalities hold true, confirming that (-frac{7}{20}) is indeed between (-frac{5}{13}) and (-frac{4}{13}).Thus, the final answer is:[boxed{-dfrac{7}{20}}]]

🔑:4. We write the fraction as frac{x}{20}. From the problem statement, we write the system of linear inequalities frac{x}{20}-7 frac{9}{13}. We are looking for an integer that lies in the intersection of both solutions of the linear inequalities. The sought integer is -7. The sought fraction is -frac{7}{20}.