Wednesday, September 29, 2010

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

If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of (p^2)*t?

1) m has more than 9 positive factors
2) m is a multiple of m^3







we already know that m is a multiple of t, so the only real issue here is whether there are 2 copies of "p" in its prime factorization.
i.e., we already know it's a multiple of pt; all that's missing is the second "p".

(1)
this clearly could be a "yes", if m is something like (p^1000)(t^1000). therefore, the challenge lies in looking for a "no".
we can get a "no" by keeping only one "p", and just raising "t" to a huge power. for instance, m = (p)(t^1000) will have over two thousand factors.
insufficient.

(2)
if m is a multiple of p^3, then it's at least a multiple of (p^3)(t), so, sufficient.

ans = (b)

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