DS Question:

If a and b are positive integers, is a a multiple of b?

1. Every prime factor of b is also a prime factor of a
2. Every factor of b is also a factor of a

Source: Princeton - Cracking the GMAT - Practise Test

Can someone answer this question and explain it?

Rephrase the Q. is b a factor of a?

from (1) lets say a = 12 and b=18

prime factors of a = 2 and 3
prime factors of b = 2 and 3

so b is not a factor

if we take b = 6 and a = 12

b is a factor so insufficient

from (2) using the same logic

b=18 gives factors = 2,3,3
a = 2 x 3 x 3 x ? which means b is a factor of a irrespective of what b is so sufficient

the above poster pretty much has the essence of this one, but here are a few additional comments.

(1)
all you have here is information about prime factors; that means you have no idea HOW MANY of those prime factors there are. the trouble here, then, is that you could easily switch the roles of a and b: this statement is true if a = 4 and b = 8 (because the only prime factor is 2), but it's also true if you switch them to a = 8 and b = 4. so that's insufficient.

(2)
here's the easiest way to figure this one:
b is a factor of itself. therefore, it has to be a factor of a.
done. isn't that awesome?
if you don't think of that, you can always try plugging different numbers; after you discover enough times in a row that b has to go into a, you should start believing that it has to happen in general.

Thanks Ron.

What got me off track was that I WAS considering the number of times a factor occurs in my evaluation (of "Every prime factor of b is also a prime factor of a") and hence options 1 and 2 appeared to be similar. Your answer clarifies this.

understandable - but, unfortunately, still incorrect. whenever you go over a problem like this, make sure you hold on to any new insights about the meaning / usage of terms.

good luck