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,
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