# Order of congruence classes

by snakesonawii
Tags: classes, congruence, order
 P: 5 1. The problem statement, all variables and given/known data If m$$\in$$Z and $$2\leq n\in Z,$$ then $$|[m]_n|=\frac{n}{(m,n)}$$ 2. Relevant equations Lagrange's Theorem 3. The attempt at a solution I am confused simply because it seems like the problem might be missing something. We are asked to find the order of the congruence class m modulo n. But I thought that to even talk about this we must first assume that m and n are coprime. Otherwise we get results like $$|[5]_{15}|=\frac{15}{5}=3$$. Yet 5^3=125 which gives you just the class 5 modulo 15 again. If we wanted to look at a cyclic group generated by $$[5]_{15}$$ we would find that it only has two elements, the classes 5 and 10 from repeated multiplication of the class 5, no inverses, and no identity (the congruence class 1 could be an identity but it is never reached by multiplication of 5 to itself).