Data Sufficiency

Can someone please tell me how to answer this question, the books explanation is shocking!!

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

(1) Every distinct prime factor of b is also a distinct prime factor of a

(2)Every factor of b is also a factor of a

Thanks in advance!

IMHO, The answer is B.

Look at it in this way.

First consider 2).
- Since every number is a factor of itself, b is a factor of b.
- It is given that every factor of b is a factor of a.

from i) and ii) b is a factor of A. Therefore A is a multiple of B.

1) is not sufficient because what about the prime factors that are repeated. two number having the same prime factors does not mean they are multiple of each other.
e.g. 12 and 18 have the same prime factor 2 and 3 but neither is the multiple of the other. But 6 and 12 have same prime factors but 12 is a multiple of 6.

Guys, please correct me if my explanation is misleading. Would it have made a difference had the the word distinct was been there ?

what if a=b
Is a still a factor of b

Sure- every number will have at least 1 and itself as a factor. So if a=b then a can be a factor of b.