When a question asks WHICH OF THE FOLLOWING MUST BE X?, try to prove that four of the answer choices DO NOT HAVE TO BE X.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
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 divisible by 25, test the SMALLEST POSSIBLE COMBINATIONS.
Case 1: Let p=2 and q=5, with the result 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, with the result that n = (5²)2 = 50.
B) q² = 2² = 4. Not a multiple of 25. Eliminate B.
The correct answer is D.
