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If x is a positive integer, is x divisible by

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If x is a positive integer, is x divisible by [#permalink]

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New post 12 Jan 2013, 05:46
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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New post 13 Jan 2013, 04:11
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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.

Answer: C.


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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New post 14 Jan 2013, 08:25
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

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Re: If x is a positive integer, is x divisible by   [#permalink] 14 Jan 2013, 08:25
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