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# If x is a positive integer, is x divisible by

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12 Jan 2013, 04:46
If $$x$$ is a positive integer, is $$x$$ divisible by 15?

(1). $$x$$ is a multiple of 10
(2). $$x^2$$ is a multiple of 12

In my opinion answer must be B. The least perfect square which is a multiple of 12 is 36 and the answer is no. The next is 144 (12^2), again no.
Explanation is not clear.

Thanks & Regards
Vinni
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Re: If x is a positive integer, is x divisible by [#permalink]

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13 Jan 2013, 03:11
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vinnik wrote:
If $$x$$ is a positive integer, is $$x$$ divisible by 15?

(1). $$x$$ is a multiple of 10
(2). $$x^2$$ is a multiple of 12

In my opinion answer must be B. The least perfect square which is a multiple of 12 is 36 and the answer is no. The next is 144 (12^2), again no.
Explanation is not clear.

Thanks & Regards
Vinni

OFFICIAL SOLUTION:

If x is a positive integer, is x divisible by 15?

(1) x is a multiple of 10 --> if x=10 then the answer is NO but if x=30 then the answer is YES. Not sufficient

(2) x^2 is a multiple of 12 --> the least perfect square which is a multiple of 12 is 36. Hence, the least value of $$x$$ is 6 and in this case the answer is NO, but if for example $$x=12*15$$ then the answer is YES. Not sufficient.

Notice that from this statement we can deduce that $$x$$ must be a multiple of 3 (else how can this prime appear in $$x^2$$?).

(1)+(2) $$x$$ is a multiple of both 10 and 3, hence it's a multiple of 30, so $$x$$ must be divisible by 15. Sufficient.

Notice that for (2) if x^2=(12*15)^2 (x^2 IS divisible by 12), then x=12*15 and in this case x IS divisible by 15.
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Re: If x is a positive integer, is x divisible by [#permalink]

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14 Jan 2013, 07:25
Bunuel wrote:

Notice that for (2) if x^2=(12*15)^2 (x^2 IS divisible by 12), then x=12*15 and in this case x IS divisible by 15.

Thanks Bunuel. I got your point.

Regards
Vinni
Re: If x is a positive integer, is x divisible by   [#permalink] 14 Jan 2013, 07:25
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# If x is a positive integer, is x divisible by

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