Show the set of congruence classes mod n is a partition of $\mathbb{Z}$.

Question

Let $[x]_{n}=${$ r\in \mathbb{Z}:r-x=nk$ for some $k\in \mathbb{Z}$} be the congruence class of x mod n. In order to show that $\left \{ [i]_{n} \right \}_{i=0}^{n-1}$ forms a partition of the set of integers, do I just show (using properties of equivalence relations) that any two congruence classes is either disjoint or equal, because that is the definition of a partition?

Answer 1

You need to demonstrate that each integer is in one partition and that no integer is in two partitions.

For the first one, consider the division lemma: for any two integers $m,n$ with $m<n$, there exist integers $k,r$ with $n=km + r$.

For the second one, what would happen if you had $a=k_1n + r_1$ and $a=k_2n + r_2$?