Proving Equivalence Classes in Modular Arithmetic

[rallycar18]

Homework Statement

Suppose [d], \in Z sub n.

 

[CompuChip]

The good news here is, that you can perform all operations in Z_m 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

 

[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.