If n is a multiple of 5 and n = p^2(q), where p and q are prime numbers, which of the following must be a multiple of 25?

a) p^2
b) q^2
c) p^2 (q^2)
d) p^2 (q)

I am confused on how to start this problem. Should I pick numbers for p and q? What is the best strategy here to ensure I can get the answer within 2 minutes?

Thanks.

hey pash 19,

I think there are a couple of ways to tackle this problem.

The easiest way, I think, would be to just remember that p^2(q) is a multiple of 5, and then move on to the answer choices.

a) p^2. This could very well be 3^2. So long as q is 5, then p^2(q) will be a multiple of 5. But p^2, when p=3, is NOT a multiple of 25. Eliminate
b) q^2. Same reasoning as above. q^2 can also be 3^2. If p=5, then p^2(q) will be a multiple of 5. But q^2, in this case is 3^2, and NOT a multiple of 25. Eliminate.
c) p^2(q^2). In this case, remember that the original p^2(q) is a multiple of 5, meaning either p or q is 5. That means that either p^2 or q^2 will be 5^2, or 25. This makes p^2(q^2) a multiple of 25. AHA!

d) p^2(q). Basic idea is the same as a) and b). p^2 can be 3^2 and q=5. 3^2(5) = 45 and that is NOT a multiple of 25. Eliminate.

I think there must have been a choice e), but the reasoning should be the same as the wrong choices.

Hope this helps.

Thank you for the great explanation! :D

you're welcome! :D