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# If the prime numbers p and t are the only prime factors of

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Manager
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If the prime numbers p and t are the only prime factors of [#permalink]

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10 Feb 2009, 09:03
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If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p^2t?

(1) m has more than 9 positive numbers
(2) m is a multiple of p^3
Senior Manager
Joined: 30 Nov 2008
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10 Feb 2009, 09:36
IMO B.

Given p and t are the only prime factors of m ==> m = $$(p^x) (t^y)$$ where x and y are the exponents of p and t and they should ateast be 1 making pt the factor of m. We need to find whether $$(p^2)*t$$ is a facotor of m

From clue 1, m has more than 9 positive factors(I am assuming factors since "more than 9 numbers" does not make any sense). We also know that p and t are the only prime factors. So The number of factors will be (x+1)(y+1) > 9. We know that x and y are atleast 1 making pt to be the factor of m. But this does not provide any information on what are the values of x and y are. Hence insufficient.

From Clue 2, m is a multiple of $$p^3$$. From the question stem, it is clear that t is also one of the prime factors. So $$p^3$$ t is a factor of m which means $$(p^2)t$$ is also a factor of m. Sufficient.
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10 Feb 2009, 10:41
nice explanation. thanks
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Re: FactorsDS   [#permalink] 10 Feb 2009, 10:41
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