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What is the smallest positive integer k such that 3528*√k is the cube

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What is the smallest positive integer k such that 3528*√k is the cube  [#permalink]

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New post 15 Aug 2019, 11:37
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What is the smallest positive integer k such that 3528*√k is the cube of a positive integer?

A. 21
B. 36
C. 441
D. 225
E. 144

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Re: What is the smallest positive integer k such that 3528*√k is the cube  [#permalink]

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New post 15 Aug 2019, 11:58
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fauji wrote:
What is the smallest positive integer k such that 3528*√k is the cube of a positive integer?

A. 21
B. 36
C. 441
D. 225
E. 144

3528 = 2^3 x 3^2 x 7^2

To make this part a perfect cube we need 3*7 = 21

Further to make √k part a perfect square we need additionally 21

Thus, the Answer must be 21*21 = 441 , (C)
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What is the smallest positive integer k such that 3528*√k is the cube  [#permalink]

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New post 15 Aug 2019, 11:59
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When we are asked to determine a number to make an expression be the square, cube, etc of a positive integer, we always have to factorize that expression and then find the number that make the exponents of all factors equal 2 (square), 3 (cube), etc.

Let's go ahead. I will apply the mentioned steps to find the answer:

1) Factorize the expression:

From the given expression, only the number 3528 can be factorized. The new expression would equal (2^3)*(3^2)*(7^2)*(k)^1/2

2) Find the number that makes the exponents of all factors equal 3 :

for the factor 2, we already have the cube of it in the expression, but for 3 and 7, we need additionally 3 and 7.

But since we are given the root of k, we would need the number to have the factors 3^2 and 7^2 so that in overall we can have (2^3)*(3^3)*(7^3)

So the right answer would be 3^2*7^2=441

Option C
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What is the smallest positive integer k such that 3528*√k is the cube   [#permalink] 15 Aug 2019, 11:59
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