I've been learning about congruence classes recently and have been having some trouble understanding the following fact; $$[a]_m = {\{b \in \mathbb{Z} : a \equiv b\pmod m}\}$$ Now, these are obviously all $b$ of the form $b = a-mk, k \in \mathbb{Z}$
Now I have trouble understanding the set of all congruence classes, $\mathbb{Z_m}$, which is defined to be ${\{[x]_m : x \in \mathbb{Z}}\}$, but my lecturer said that if $m$ is fixed, then this set is equivalent to the set ${\{0,1,2,\dots,m-1}\}.$ How is this true when $b$ can be of the form $a-mk$ for any $k \in \mathbb{Z}?$
Each congruence class has exactly $1$ element in $\bigl[\mkern-4.2mu\lvert0\cdot\cdot\: n-1|\mkern-4.2mu\bigr] $ (the remainder of the Euclidean division by $m$. Thus the set of congruence classes modulo $m$ is in bijection with this set of representatives.