GMAT Math

If p and q are positive integers, is every factor of p a factor of q?

(1) p is a factor of q.

(2) The product of the prime factors of p is a factor of q.

whats the OA of this ?

It should be either A or D.

oki

(1) p is a factor of q. Since p is a factor of q (q/p=integer), then obviously every factor of p will be a factor of q. Sufficient. (Take example like p = 6 and q = 36)

(2) The product of the prime factors of p is a factor of q. If p=36 and q=6, then the product of the prime factors of p (2*3=6) IS a factor of q but not every factor of p is a factor of q (for example 36 is not a factor of 6). Though if p=q=6, then we'll have that every factor of p a factor of q. Not sufficient.

Answer: Must be A.

St1: If P is a factor of Q, every factor of P will be a factor of Q.
lets say P = 6 and Q = 12
factors of P = {1,2,3,6}
factors of Q = {1,2,3,4,6,12}
Every factor of 6 must be a factor of 12
Hence sufficient

St2: Product of prime factors of p is a factor of q.
if p = 36, its factors are {1,2,3,4,6,9,12,18,36} of which prime factors are 2 and 3. their product = 6.
q must be a multiple of 6.lets say q = 6 then 2 and 3 are factors of both (p and q) but several others factors of p like 9, 12 etc are not factors of q.
Alternatively if q = 72, then every factor of p will be a factor of q.
hence insufficient

Ans A

Nicely done by srcc25anu. The answer is A.