# Equivalence class problem

1. Jun 9, 2010

### rallycar18

1. The problem statement, all variables and given/known data

Suppose [d], $$\in$$ Z sub n.

Last edited: Jun 9, 2010
2. Jun 9, 2010

### CompuChip

The good news here is, that you can perform all operations in Zm by picking any representative and working in the integers.

So if c sits in the conjugacy class of a, you can write
c = a + i m
where 0 <= a < m and i is some integer, similarly
d = b + j m

3. Jun 9, 2010

### CompuChip

If cd is congruent to ab mod m, then cd lies in [ab] by definition, doesn't it?
If you want a less trivial proof, show that
cd = ab + km
for some integer k.