WishMasterUA wrote:

if n is a multiple of 5 and n=p^2q, where p and q are prime number, which of the following must be multiple of 25?

1) p^2

2) q^2

3) pq

4) p^2q^2

5) p^3q

A common phrase that is used on the GMAT is the word must. In this question, we are asked which of the following must be a multiple of 25. This means that one of our answer choices will always be a multiple of 25, no matter what. It is our job to determine which one, based on the given information.

We are given that n is a multiple of 5, n = (p^2)q, and that p and q are prime numbers.

Because n is a multiple of 5, a prime number, we know that either p or q is 5. Let’s now analyze each answer choice to determine which one MUST (in all cases) be a multiple of 25.

A) p^2

If p = 3, then p^2 = 9 is not a multiple of 25. Answer choice A is not correct.

B) q^2

If q = 3, then q^2 = 9 is not a multiple of 25. Answer choice B is not correct.

C) pq

If p = 5 and q = 3 (or vice versa), pq = 15 is not a multiple of 25. Answer choice C is not correct.

D) (p^2)(q^2)

Regardless of which values we select for p and q, since we know that either p or q is 5, (p^2)(q^2) will always be a multiple of 25. If this is difficult to see, let’s use numbers.

If p = 5 and q = 3, (p^2)(q^2) = (25)(9) is a multiple of 25.

If p = 3 and q = 5, (p^2)(q^2) = (9)(25) is also a multiple of 25.

Answer choice D is correct.

For practice, let’s analyze answer choice E.

E) (p^3)q

If p = 3 and q = 5, then (p^3)q = 135 is not a multiple of 25. Answer choice E is not correct.

Answer: D

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