Problem:
Find the number of positive integers that are divisors of at least one of , 18^
Solution:
Note that , so it has divisors. Similarly, , so it has divisors, and , so it has divisors. There are divisors of both and , namely those numbers that are divisors of ; there are divisors of both and , namely those numbers that are divisors of ; and there are divisors of both and , namely those numbers that are divisors of . There is only one divisor of all three. Therefore, the Inclusion-Exclusion Principle implies that the number of divisors of at least one of the numbers is .
The problems on this page are the property of the MAA's American Mathematics Competitions