Problem Solving

If n is 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) p³q

If p and q are prime numbers, and p²q is divisible by 5, then either p = 5, q = 5 or they both equal 5.

We're looking for an expression that MUST be divisible by 25, which means there must be TWO 5's "hiding" in the expression.

A) p²
We cannot be certain that there are TWO 5's "hiding" in this expression.
It could be the case that p = 2 and q = 5, in which case p² is NOT divisible by 25
ELIMINATE A

B) q²
We cannot be certain that there are TWO 5's "hiding" in this expression.
It could be the case that p = 5 and q = 2, in which case q² is NOT divisible by 25
ELIMINATE B

C) pq
We cannot be certain that there are TWO 5's "hiding" in this expression.
It could be the case that p = 5 and q = 2, in which case pq is NOT divisible by 25
ELIMINATE C

D) p²q²
YES, we can be certain that there are TWO 5's "hiding" in this expression.
If p = 5, then p²q² = 25q², which is DEFINITELY divisible by 25
If q = 5, then p²q² = 25p², which is DEFINITELY divisible by 25

E) p³q
We cannot be certain that there are TWO 5's "hiding" in this expression.
It could be the case that p = 2 and q = 5, in which case p³q is NOT divisible by 25
ELIMINATE E

Answer = D

Cheers,
Brent