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is m a multiple of t*p^2

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Source: — Data Sufficiency |

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by Anurag@Gurome » Wed Dec 07, 2011 5:01 am
jainrahul1985 wrote:If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of t*p^2?
(1) m has more than 9 positive factors
(2) m is a multiple of p^3

OA B
Let m = (p^x)*(t^y). Then m has (x + 1)(y + 1) positive factors.

(1) m has more than 9 positive factors implies that (x + 1)(y + 1) > 9
If x = 2, y = 3, then m = p² * t^3 = t * p² * t², which is clearly a multiple of t * p².
If x = 1, y = 4, then m = p * t^4, which is not a multiple of t * p².
No definite answer; NOT sufficient.

(2) m is a multiple of p^3 implies m = k * p^3, which implies k has to be t^a * p^b (where a ≥ 1 and b ≥ 0)
We know that m = p^x * t^y and m = t^x * p^y * p^3) will be a multiple of (p^2)*t; SUFFICIENT.

The correct answer is B.
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by GMATGuruNY » Wed Dec 07, 2011 6:07 am
jainrahul1985 wrote:If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of t*p^2?
(1) m has more than 9 positive factors
(2) m is a multiple of p^3

OA B
I received a PM, so here's another approach.

We know that m is a multiple of tp, since t and p are factors of m.
In order to determine whether m is a multiple of tp², we need to know whether m is a multiple of p².

Question rephrased: Is m a multiple of p²?

Statement 1: m has more than 9 positive factors.
If m = tp¹�� -- yielding far more than 9 factors -- then m is a multiple of p².
If m = t¹��p -- yielding far more than 9 factors -- then m is not a multiple of p².
INSUFFICIENT.

Statement 2: m is a multiple of p³.
Thus, m is a multiple of p².
SUFFICIENT.

The correct answer is B.
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