ayankm wrote:123) If n is multiple of 5, and n = p^2 * q where p and q are prime, which of the following must be a multiple of 25?

a. p^2
b. q^2
c. p*q
d. p^2 * q^2
e. p^3 * q

Not getting a clue how to go about with this.
Any help or clue will be much appreciated.

Hi, since n is a multiple of 5, and since p and q are both prime, then we know that at least one of p or q must equal 5.

So, let's rephrase the question:

if either p or q = 5, which of the following MUST be a multiple of 25?

a) p^2... if q=5, p^2 could be anything... not a MUST.

b) q^2... if p=5, q^2 could be anything... not a MUST.

c) p*q... must be a multiple of 5, but doesn't have to be a multiple of 25.

d) p^2 * q^2... since either p or q = 5, either p^2 or q^2 must be 25; accordingly, p^2*q^2 = 25*(integer) which MUST be a multiple of 25... correct!

(On test day we'd stop there, but let's go into review mode and look at (e).)

e) p^3 * q... if q=5, then p^3*q must be a multiple of 5, but not of 25... not a MUST.

Hello ayankm,

(p^2)*q is a multiple of 5 so either p is 5, q is 5 or they are both equal to 5 (since they are each prime). Which of the following must be a multiple of 25 (a multiple of 5*5)?

A) not a multiple of 25 if p is not equal to 5.
B) not a multiple of 25 if q is not equal to 5.
C) pq is a multiple of 5, but it will only be a multiple of 25 if both p and q equal 5.
D) must be a multiple of 25 because if p is 5, then p^2 is 25. If q is 5, then q^2 is 25. Either way, the product will be a multiple of 25.
E) definitely a multiple of 5, but if p is not equal to 5, then the product will not be a multiple of 25.

The answer is D. You can study a more detailed solution or watch a step-by-step video solution as this is GMATPrep question 1236. To practice similar questions, set topic='number properties' and difficulty='600-700' in the drill engine.

Hope that makes sense,
-Patrick