There’s a neat trick for finding the number of factors of a number. First find the prime factorization of the number, for example:

Add to the power of each prime factor and then multiply those numbers together. In this example,

factors.

This reflects the fact that each factor of contains where , or and , where , or and where or .

I used this property to solve a problem in a number theory class I’m taking. The problem asks to compute the sum of all positive integers such that has positive factors.

Since , working backward we are looking for a prime factorization with 2 primes raised to the powers of 2 and 6. The prime factorization of so we have one prime raised to the sixth power. All we need is the other prime to be squared. Setting gives us

with divisors.