1. The problem statement, all variables and given/known data Prove that if gcd(a, 133) = 1, then 133 divides (a^18 - 1). 3. The attempt at a solution This is an old homework question as I'm going over the homeworks to review for the test, but can't seem to get this right. Which is annoying because I remember I did it fine back in the course when this homework was actually due. So, I know gcd(a, 133) = 1 implies 133 divides a^phi(133) -1 = a^108 -1. So I assume I'd want to factor that out so that I get a product of polynomials, one of the form (a^18 - 1), and show that the other polynomial factors aren't divisible by 133 by appealing to their modulus form. But that doesn't seem to be working out.