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st1: insuff, for example m=9 has only one prime factor. 18 has 2.
st2: insuff, for example m=18 (then m/3=6 has two prime factors) has 2 prime factors, and m=30 (m/3=10 has two prime factors) has 3 prime factors.

together they are sufficient:
st2 says that m/3 has 2 prime factors. so m/3=p^x*q^y where p and q are primes.
st1 claims that one of the two factors found in st2 is 3. or in other words
m/3=3^x*q^y (q is prime, x and y are integers)

from this we can infer: m = 3^(x+1)*q^y
so m has exactly 2 factors.

hence C is the answer....

Last edited by hobbit on 05 Jan 2007, 07:50, edited 1 time in total.

I always get confused in these kinda problems? I cannot figure out whether the question is asking for all different prime factors or just all prime factors. I think the answer is C if the question is asking for all different prime factors, but E if its asking for all prime factors, including repeats.
Any explanation would be helpful.

I always get confused in these kinda problems? I cannot figure out whether the question is asking for all different prime factors or just all prime factors. I think the answer is C if the question is asking for all different prime factors, but E if its asking for all prime factors, including repeats. Any explanation would be helpful.

in the set of all factors there are no repeats, and so is the set of prime factors.
so 4 has 3 factos 1,2,4 and only one prime factor (2)
it is true that 4 can be divided twice to 2... the decomposition of 4 to multiples of prime numbers has 2 twice.... but you must differentiate between decomposition of a number into primes and the prime factors of a number.... these are two different things.