I have a convergent sum where I use the reciprocal of a_n at each step: a_n = a_n / gcd(a_n, b_n) <--- I'm removing common factors. This converges as long as I want to run it. Both a_n and b_n are quite dynamic. However, if a_n equals b_n then the divide after the gcd would return 1 and the series would diverge. Is there a way to prove that a_n and b_n will never be equal? I can match this series with another that uses the same method but it has been proven that the divide and gcd will never return a 1. Can I use that first series to prove that the second series will never have the divide and gcd return 1? Are there any theorems that deal with this condition? Any help is appreciated.