is m a multiple of t*p^2

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jainrahul1985: Master | Next Rank: 500 Posts; Posts: 228; Joined: Sun Aug 17, 2008 8:08 am; Thanked: 4 times

by jainrahul1985 » Wed Dec 07, 2011 3:23 am

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

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Source: Beat The GMAT — Data Sufficiency | Wed Dec 07, 2011 3:23 am

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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 = pÂ² * t^3 = t * pÂ² * tÂ², which is clearly a multiple of t * pÂ².
If x = 1, y = 4, then m = p * t^4, which is not a multiple of t * pÂ².
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 â‰¥ 1 and b â‰¥ 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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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

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 tpÂ², we need to know whether m is a multiple of pÂ².

Question rephrased: Is m a multiple of pÂ²?

Statement 1: m has more than 9 positive factors.
If m = tpÂ¹â�°â�° -- yielding far more than 9 factors -- then m is a multiple of pÂ².
If m = tÂ¹â�°â�°p -- yielding far more than 9 factors -- then m is not a multiple of pÂ².
INSUFFICIENT.

Statement 2: m is a multiple of pÂ³.
Thus, m is a multiple of pÂ².
SUFFICIENT.

The correct answer is B.

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