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# What is the smallest positive integer k such that

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Intern
Joined: 21 Dec 2015
Posts: 12
What is the smallest positive integer k such that  [#permalink]

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28 Jun 2016, 01:19
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Difficulty:

45% (medium)

Question Stats:

54% (01:25) correct 46% (01:23) wrong based on 62 sessions

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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
Current Student
Joined: 22 Jun 2016
Posts: 245
Re: What is the smallest positive integer k such that  [#permalink]

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28 Jun 2016, 01:42
1
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.

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Math Expert
Joined: 02 Sep 2009
Posts: 52231
What is the smallest positive integer k such that  [#permalink]

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28 Jun 2016, 04:00
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

$$126=2*3^2*7$$, so in order $$126*\sqrt{k}$$ to be a square of an integer $$\sqrt{k}$$ must complete the powers of 2 and 7 to even number, so the least value of $$\sqrt{k}$$ must equal to 2*7=14, which makes the leas value of $$k$$ equal to 14^2=196.

This question is discussed here: http://gmatclub.com/forum/baker-s-dozen ... l#p1057505
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Joined: 31 Jan 2018
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Re: What is the smallest positive integer k such that  [#permalink]

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03 Feb 2018, 21:22
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
Math Expert
Joined: 02 Aug 2009
Posts: 7199
Re: What is the smallest positive integer k such that  [#permalink]

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03 Feb 2018, 22:06
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 $$\sqrt{k}=2*7.....k=2^2*7^2=14^2=196$$
D
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: What is the smallest positive integer k such that  [#permalink]

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03 Feb 2018, 22:12
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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Joined: 02 Sep 2009
Posts: 52231
Re: What is the smallest positive integer k such that  [#permalink]

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03 Feb 2018, 23:00
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

Merging topics. Please search before posting. Thank you.
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Re: What is the smallest positive integer k such that &nbs [#permalink] 03 Feb 2018, 23:00
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