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.
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