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题库 / Prep2007E2-DS-166
If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p2t ?
(1) m has more than 9 positive factors.
(2) m is a multiple of p3.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
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1) m has more than 9 positive factors
INSUFFICIENT: it doesnt tell exponent/powers of prime factors p & t. We dont know whether m is multiple of p^2.
2) m is a multiple of p3p3
SUFFICIENT: If m is a multiple of p3p3, then m must be multiple of p2p2. As 't' is also a prime factor of m, then m must be multiple of p2∗tp2∗t
e.g. say m=24, p=2, t=3. As 24 is multiple of p3=23=8p3=23=8, 24 must be multiple of p2=22=4p2=22=4, and therefore 24 is also multiple of p2∗t=22∗3=6
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