Factor relashionship

[Xbond]

Hi there,

Could you help me to resolve this PS and explain the step of resolution

If n is a multiple of 5 and n=p²q, where p and q are prime numbers, which of the following must be a multiple of 25 ?

A) p²
B) q²
C) pq
D) p²q²
E) p²q

[cbenk121]

Ok, so we know p^2*q is divisible by 5.

Suppose p = 3, and q = 5. That would equal 45. Disproves (A), (C), and (E).

Now suppose p = 5 and q = 3. That would equal 75. This disproves (B).

However, D has held true in both scenarios, so that's the answer.

This makes sense, in hindsight of course. To be divisible by 5, there must be a factor of five. Either q or p equals 5, since we are given that q and p are prime.

However, both do not have to equal 5, so therefore the only way to ensure that 25 is a factor of the resulting number is to square BOTH p and q, as 5x5 is 25.

[bikerguy.gmat]

POE is the best method for this.

Since the questions says "which of the following must be ...", it means that the answer options dont have a choice but to be true for any random legitimate value (i.e. no conditions on what values to choose).

try plugging in random values for p and q to satisfy the original equation, and keep eliminating incorrect ans. options.

i hope this helps - please let me know if further explanation required.

[Xbond]

many thks
oa is D