2021 JMC 10 Problems/Problem 25
Problem
How many ordered pairs of positive integers with and exist such that neither the numerator nor denominator of the below fraction, when completely simplified (i.e. numerator and denominator are relatively prime), are divisible by five?
Solution
The problem asks for when and have the same number of powers of 5.
First, when is even, the numerator is not divisible by , so the denominator must also not be divisible by . So if and only if . This gives solutions.
When is odd, has at least 3 powers of due to the sum of -th powers factorization for odd . In order for , we must have . Note that (taking and/or modulo 125 will work). Therefore, . It is clear that must be satisfied. To verify that all with work, we check with Lift-the-Exponent Lemma; where is the maximum possible such that divides . So when is odd, there are solutions. Thus, we have in total, solutions.