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# Is x the square of an integer?

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Manager
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Is x the square of an integer? [#permalink]

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24 Apr 2010, 22:25
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Is x the square of an integer?

(1) x = 12k + 6, where k is a positive integer
(2) x = 3q + 9, where q is a positive integer

OPEN DISCUSSION OF THIS QUESTION IS HERE: is-x-the-square-of-an-integer-126385.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 13 Nov 2013, 02:12, edited 1 time in total.
Edited the question and added the OA.
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Re: Is x the square of an integer [#permalink]

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25 Apr 2010, 00:30
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From Stmt 1 ,
x= 6(2k+1) where k = positive integer.
2k + 1 is always odd.
To be a square of an integer, 2k+1 has to be a multiple of 6 and that is not possible.
So, x can't be square of an integer.

Stmt 1 sufficient.

From Stmt 2,
x= 3(q+3)
putting different values of q can have 2 different results.
Example put q=1, x =12 ... not a square of an integer
put q=24 , x = 81 ... square of an integer 9.

Stmt 2 ,Not sufficient.

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Re: Is x the square of an integer? (1) x = 12k+6 , where k is a [#permalink]

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13 Nov 2013, 01:38
Bunuel is their some expert easy way to solve it?
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Like my post Send me a Kudos It is a Good manner.
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Re: Is x the square of an integer? [#permalink]

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13 Nov 2013, 02:13
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Expert's post
Is x the square of an integer?

(1) x = 12k + 6, where k is a positive integer --> $$x=6(2k+1)=2*3(2k+1)$$. Now, $$x$$ to be a perfect square it should have an even power of its primes, but $$2k+1$$ is an odd number and can no way produce 2 for $$x$$. Thus $$x$$ is not a perfect square. Sufficient.

(2) x = 3q + 9, where q is a positive integer --> $$x=3(q+3)$$. If $$q=1$$ then $$x=12$$ and the answer is NO but if $$q=9$$ then $$x=36$$ and the answer is YES (basically if (q+3)=3*any perfect square then x will be a perfect square and if (q+3) is some other type of number then x won't be a perfect square). Not sufficient.

OPEN DISCUSSION OF THIS QUESTION IS HERE: is-x-the-square-of-an-integer-126385.html
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Re: Is x the square of an integer?   [#permalink] 13 Nov 2013, 02:13
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# Is x the square of an integer?

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