Hopefully I can be succinct here, lets see.

If n is a multiple of 5, then n can be expressed as 5x where x is an integer.

If I substitute in the following eqn, I get 5x = p^2 * q

Now I know that, since x is an integer, that:

{1} (p^2 * q)/5 is also an integer.

If {1} is an integer and p and q are primes, the only way this can be an integer is if either p is 5 or q is 5. That leaves two things to consider, (1) p is 5 and q is some other prime number OR (2) p is some other prime and q is 5.

Now I'll look at each answer choice under each scenario side-by-side. The correct answer choice will be the one that is divisible by 25 under either circumstance (clue is "which answer choice MUST be divisible by 25).

Answer choice A, (1) 25; (2) q^2
Answer choice B, (1) p^2; (2) 25
Answer choice C, (1) 5q; (2) p5
Answer choice D, (1) 25q^2; (2) (p^2)25
Answer choice E, (1) 125q; (2) (p^3)5

As you can see, only Answer choice D offers both parts being divisible by 25.

Good luck, I have a feeling that I'll have to explain this one.

Hi Ryan, I was wondering why we didn't consider both p and q each being equal to 5?

P and q each equal to 5 is a possibility but in this case A B C D and E are right.
The important word in this type of question is MUST, so whatever the possibilities we have, our answer must be good.

Looks like pepeprepa beat me to it. Can I provide further explanation?

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

A. p^2
B. q^2
C. pq
D. p^2q^2
E. p^3q
I got this correct as well but need some light to completely understand this question..
Thanks.

here is my solution:-
Since n is a multiple of 5, n^2 must be a multiple of 25(you can test it by plugging in values).
Since n=p^2q, Square both sides n^2=(p^2q)^2
Hence (p^2q)^2 is the solution

Sunny22uk wrote:

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

A. p^2
B. q^2
C. pq
D. p^2q^2
E. p^3q
I got this correct as well but need some light to completely understand this question..
Thanks.

here is my solution:-
Since n is a multiple of 5, n^2 must be a multiple of 25(you can test it by plugging in values).
Since n=p^2q, Square both sides n^2=(p^2q)^2
Hence (p^2q)^2 is the solution

(p^2q)^2 = (p^4)(q^2)

which is not in the answer list. what am i missing? thanks

could anyone please explain? Thanks!

Here's how i did it.

n = p^2*q, since n is multiple of 5, RHS has to be multiple of 5. There are 2 ways that can happen

1. p = 5 and q = anything (including 5)
2. q = 5 and p = anything (including 5)

run through these options, starting in the middle and you'll get the right answer as p^2*q^2, once again the word MUST in the question is important.