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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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转自Abbigail_law
If the prime numbers p and t are the only prime factors of integer m
=> m=(p^a)(t^b) 且a>=1, b>=1
is m a multiple of p^2t? => 即要求证a>=2
1) m has more than 9 positive factors => (a+1)(b+1)>9 不能保证a>=2 比如a为1，b为4就不行
2) m is a multiple of p^3 ＝> 直接给出 a>=3
任何一个自然数的因子个数 为质因数分解中 每个质因子的指数加1的乘积
i.e. 20=(2^2)(5^1) 所以20的因子个数为（2＋1）（1＋1）＝6个，分别是1，2，4，5，10，20
所以B.
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