r, s > 0, integers

"Is r/s an integer?"
= "Is r a multiple of s?"
= "Is r divisible by s?"
= "Is s a factor of r?"

Statement 1) If every factor of s is a factor of r, then s is a factor of r.

For example, if s = (5)(6) = 30 and r = (30)(10)(5)(6) = 9000

r/s = (30)(10) = 300, which is an integer. Sufficient.

Statement 2) If every prime factor of s is a prime factor of r, then every factor of s is a factor of r, and s is a factor of r.

Every positive integer can be reduced to a factorization consisting only of prime numbers. From the above example we would have:

s = (2)(3)(5) = 30 and r = (2^3)(3^2)(5^3) = (2)(3)(5)(2^2)(3)(5^2)

s/r = (2^2)(3)(5^2) = 300, which is an integer. Sufficient.

D

This is from a GMAT Prep practice test from the GMAT website, and this was the answer that I thought as well, however, the test said D was wrong and the correct answer was A......I just wanted to make sure that I wasn't crazy and that the test had an incorrect answer, thank you for your post.

For A to be correct, I would have thought it would be worded "every unique prime factor of s is also a unique prime factor of r" or "every different prime factor..."

But, I misinterpreted the statement, because the factors for a number, say 18, are all possible combinations of its prime factors, 2^1 * 3^2: 1, [2], [3], 6, 9, 18. The prime numbers only ever appear here once.

Thus:

Statement 2) If every prime factor of s is also a prime factor of r,

s could be 2^100 * 3^100 * 7^100
and r = 2 * 3 * 7

They would share the same prime factors:
r: 1, [2], [3], 6, [7], 14, 21, 42
s: 1, [2], [3], 4, 6, [7], 8, 9, 12, ...


r/s = 1/(2^99) * 1/(3^99) * 1/(7^99), definitely not an integer.

The r/s is supposed to be r divided by s, if that makes a difference, looking at your explanation that might be a point of confusion.....