Problem:
Find the least positive integer such that is a product of at least four not necessarily distinct primes.
Solution:
Considering the expression modulo and shows that none of these primes can be a factor of . Thus the smallest possible value for with four prime factors is , but there is no integer for which . The next candidate for the smallest value for with four prime factors is . If there actually is an integer such that , then because is a multiple of , there must be an integer with either or . If , the equation becomes . It follows that must be divisible by . Clearly does not work, but satisfies the equation. If , there are no small values of that satisfy the needed condition. Therefore the least positive integer satisfying the needed condition is .
The problems on this page are the property of the MAA's American Mathematics Competitions