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What is the smallest positive integer k such that

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Rushilkhera
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What is the smallest positive integer k such that [#permalink] New post   28 Jun 2016, 02:19
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What is the smallest positive integer K such that \(126*\sqrt{k}\) is the square of a positive integer?

A. 14
B. 36
C. 144
D. 196
E. 441

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Re: What is the smallest positive integer k such that [#permalink] New post   28 Jun 2016, 02:42
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126= 2*3*3*7

So, for 126(sqrt(k)) to be the square of a positive integer, we need all the factors of 126 to have one pair. 3 already has one pair, so pair for 2 and 7 is required.

Therefore, sqrt(k) should be 2*7 ---> k = (14)^2 = 196.

Hence, answer is D.

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Re: What is the smallest positive integer k such that [#permalink] New post   28 Jun 2016, 05:00
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Rushilkhera wrote:
What is the smallest positive integer K such that \(126*(k)^1^/^2\) is the square of a positive integer?

A. 14
B. 36
C. 144
D. 196
E. 441


\(126=23^27\), so for \(126*\sqrt{k}\) to be the square of an integer, \(\sqrt{k}\) must complete the powers of 2 and 7 to an even number. Therefore, the smallest value of \(\sqrt{k}\) must be \(2*7=14\), which makes the smallest value of \(k\) equal to \(14^2=196\).

Answer: D.


This question is discussed here: https://gmatclub.com/forum/baker-s-doze ... l#p1057505
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Re: What is the smallest positive integer k such that [#permalink] New post   03 Feb 2018, 23:06
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ggpapas wrote:
What is the smallest positive integer k such that \(126*(k)^{1/2}\) is the square of a positive integer?
A.14
B.36
C.144
D.196
E.441


\(126*\sqrt{k}=2*7*3^2*\sqrt{k}\)
So \(\sqrt{k}=2*7.....k=2^2*7^2=14^2=196\)
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Re: What is the smallest positive integer k such that [#permalink] New post   03 Feb 2018, 23:12
Lets start by factoring 126 first, we get , \(126= 2 * 3^2 * 7\)

So for 126 to be square we need one 2 and one 7, which will be provided by the min value of K ( note we already have two 3's hence dont require additional 3 )

Now lets ask minimum value of k to give one 2 and one 7 ===> k= \(2 * 7\)

Note : k already is raised to the power \frac{1}{2} So k can only give us one 2 and one 7, when min k itself has two 2's and two 7's

hence min k needs to have = 2^2 * 7^7

so min k= 196
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Re: What is the smallest positive integer k such that [#permalink]
03 Feb 2018, 23:12
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