Problem:
Suppose that is the product of three consecutive integers and that is divisible by . Which of the following is not necessarily a divisor of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
In any triple of consecutive integers, at least one is even and one is a multiple of . Therefore, the product of the three integers is both even and a multiple of . Since is adivisor of the product, the numbers , and must also be divisors of the product. However, contains two factors of , and need not. For example, is divisible by , but not by .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions