If P and Q are positive integers, are P and Q equal?

If P and Q are positive integers, are P and Q equal?

(1) LCM of P and Q = GCD of P and Q
(2) P and Q have same prime factors

GMAT Track of thought 1
P can be anything from the set {1,2,3,.....} and Q can also be anything from the set {1,2,3,.....}.
Is P=Q? I can have a definite YES or a definite NO as an Answer.

GMAT Track of thought 2
(1) LCM of P and Q = GCD of P and Q
GCD is < LCM and at the most its equal when the numbers are equal.
So P=Q. Sufficient. Eliminate B,C,E. We are down to two options A , D.

GMAT Track of thought 3
(2) P and Q have same prime factors
This is clearly insufficient because I am not sure what the powers of the prime factors are!!
For example P can be 2 *5 and Q can be 20( Since 20=2^2 * 5 with the prime factors 2 and 5 like P )
Does this mean P=Q? NO
But what if I take P= 2*3 and Q=2*3 with the same prime factors 2 and 3.
No P=Q and the answer to the question stem is YES.
Insufficient.
Eliminate D.

Hence, the correct answer is option A.
Hope this is clear.
Let me know if you have any questions on this.

Devmitra Sen
GMAT Mentor