Problem:
How many positive integer multiples of can be expressed in the form , where and are integers and ?
Solution:
Because
and is relatively prime to , it is necessary to find and so that is divisible by the primes and . Notice that is the smallest power of that leaves a remainder of when divided by or , and that is the smallest power of that leaves a remainder of when divided by . Hence is divisible by if and only if for some positive integer . Thus it is necessary to count the number of integer solutions to
For each , there are suitable values of (and ), so the number of solutions is
The problems on this page are the property of the MAA's American Mathematics Competitions