1. The problem statement, all variables and given/known data For what values of a (mod 78) will ax ≡ 26 (mod 78) have exactly 13 solutions? 2. Relevant equations gcd, etc. 3. The attempt at a solution A solutions exists if gcd(a,n) | b. gcd(a, 78) | b Let d = gcd(a, n). If d|b, then ax ≡ b (mod n) has exactly d solutions. I want to set d = 13 = gcd(a, 78) and 13 | 78. I know this is a simple problem, but I'm obviously missing something.