Problem Solving

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

n = p^2q
n is a multiple of 5 and p is a prime => p must be 5
=>p^2 is 25
IMO "A"
But option "D" is also a possible one as p^2q^2 is a factor of p^2, ie.. 25

fangtray 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

When a question asks for WHAT MUST BE X, try to prove that four of the answer choices DO NOT HAVE TO BE X.
The correct answer will be the remaining answer choice.

In order for n to be a multiple of 5, either p and/or q must be a multiple of 5.
Since the goal is to prove that four of the answer choices do NOT have to be a multiple of 25, start with the SMALLEST POSSIBLE COMBINATIONS.

Case 1: Let p=2 and q=5, so that n = 2²(5) = 20.
A) p² = 2² = 4. Not a multiple of 25. Eliminate A.
B) q² = 5² = 25. 25 is a multiple of 25. Hold onto B.
C) pq = 2*5 = 10. Not a multiple of 25. Eliminate C.
D) p²q² = 2²(5²) = 100. 25 is a multiple of 25. Hold onto D.
E) p³q = (2³)5 = 40. Not a multiple of 25. Eliminate E.

Case 2: Let p=5 and q=2, so that n = (5²)2 = 50.
B) q² = 2² = 4. Not a multiple of 25. Eliminate B.

The correct answer is D.

GMATGuruNY wrote:

fangtray 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

When a question asks for WHAT MUST BE X, try to prove that four of the answer choices DO NOT HAVE TO BE X.
The correct answer will be the remaining answer choice.

In order for n to be a multiple of 5, either p and/or q must be a multiple of 5.
Since the goal is to prove that four of the answer choices do NOT have to be a multiple of 25, start with the SMALLEST POSSIBLE COMBINATIONS.

Case 1: Let p=2 and q=5, so that n = 2²(5) = 20.
A) p² = 2² = 4. Not a multiple of 25. Eliminate A.
B) q² = 5² = 25. 25 is a multiple of 25. Hold onto B.
C) pq = 2*5 = 10. Not a multiple of 25. Eliminate C.
D) p²q² = 2²(5²) = 100. 25 is a multiple of 25. Hold onto D.
E) p³q = (2³)5 = 40. Not a multiple of 25. Eliminate E.

Case 2: Let p=5 and q=2, so that n = (5²)2 = 50.
B) q² = 2² = 4. Not a multiple of 25. Eliminate B.

The correct answer is D.

thanks ! this clears it up. can we derive anything from the fact that 5*x = p*p*q?

I thought p^2q as p^(2q). That's why there was ambiguity in my explanation. It's better to represent it as qp^2.

To mathbyvemuri,
Its my bad for not better writing out the question - I did write this super quickly before my coworkers got into work. Thakns for posting the answer thought!