Problem:
It is known that, for all positive integers ,
Find the smallest positive integer such that is a multiple of .
Solution:
The sum is a multiple of if and only if for some positive integer . Because is odd and and cannot both be even, it follows that either or is a multiple of . Furthermore, the product is divisible by for all integer values of . (Why?) Substitute , and check whether is divisible by to see that is the smallest positive integer for which is a multiple of .
The problems on this page are the property of the MAA's American Mathematics Competitions