order of congruence classes


by snakesonawii
Tags: classes, congruence, order
snakesonawii
snakesonawii is offline
#1
Apr12-11, 04:14 PM
P: 5
1. The problem statement, all variables and given/known data

If m[tex]\in[/tex]Z and [tex]2\leq n\in Z,[/tex] then [tex]|[m]_n|=\frac{n}{(m,n)}[/tex]

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 [tex]|[5]_{15}|=\frac{15}{5}=3[/tex]. 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 [tex][5]_{15}[/tex] 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).
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