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

Draw a prime box and put p and t inside. According to the problem, there could be multiple instances of p and t in there, but that's it. We want to know whether there are at least two p's and one t in there.

Start with statement 2. If m is a multiple of p^3, that means there are 3 p's in m's prime box. There's already a t in there, according to the original question. So there are at least 2 p's and one t. Answer to question is yes, so statement is sufficient. Eliminate A, C, E.

Statement 1. Notice that this just says "positive factors" NOT prime factors. The complete set of factors is made by multiplying the prime factors in different combinations. For example, 12 has the prime factors 2, 2, and 3. We can find all of the general factors of 12 by taking 2, 3, 2*2, 2*3, 2*2*3, and of course 1.

So m has more than 9 positive factors. Well, I know m has p and t - there are 2 factors. And I know m has 1 and itself - there are 2 more factors, for a total of four. I need five more, so I have to add to my prime box to be able to create five more general factors. The only things I can put in my prime box are p and t. I can put all p's, all t's, or some combination of p's and t's. If I put in at least one p, then I'd have at least 2 p's and one t, which would answer the question "yes." BUT, if I put in all t's, then I'd only have one p, which would answer the question "no" - so the statement is insufficient.

Hence B.

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Thanks & Regards,

Anaira Mitch