jose.mario.amaya wrote:p^a* q^b* r^c* s^d = x, where x is a perfect square. If p, q, r, and s are prime integers, are they distinct?

(1) 18 is a factor of ab and cd

(2) 4 is not a factor of ab and cd

Dear jose.mario.amaya:
I'm happy to help with this.

In any perfect square, all the prime factors must have even powers. If p, q, r, and s are distinct prime numbers, than a, b, c, and d would each have to be event. If one of those exponents is odd --- say, there were a 3^5 as one of the factors, then we would need another 3^(some odd number), so that overall 3 had an even exponent. If all the prime factors are distinct, then the four exponents have to be even. If the four exponents are even, the prime factor may or may not be distinct. If some of the four exponents are odd, then four prime factors definitely are not all distinct. (I have edited the statements for greater clarity.)

Statement #1: 18 is a factor of both ab and cd
Well, this doesn't guaranteed that a & b & c & d are all even. They could all be even, but they don't have to be. We could have a = b = c = d = 6, or a = c = 6 and b = d = 3 --- all even, or some even and some odd. Even if they are all even, this is consistent with the condition of all distinct prime factors, but it doesn't guarantee that the prime factors are distinct. We can draw no conclusion on the basis of this statement. This is insufficient.

Statement #2:4 is a factor neither of ab nor of cd
Well, an even number times an even number would have to be divisible by 4, so if ab is not divisible by 4, at least one of the numbers must be odd. If one of the exponents is odd, then we know the prime factors are not distinct. On the basis of this statement, we can draw a firm conclusion. This is sufficient.

Answer = B

Does all this make sense?
Mike