Equivalent expressions for quotient set $(\mathbb{Z},+)/(m\mathbb{Z},+)$

Question

Inspired by this answer I tried to write down the quotient set $(\mathbb{Z},+)/(m \mathbb{Z},+)$ with $m \in \mathbb{N}$. Now If I got it right, it should be $$ \{ m\mathbb{Z}+0, m\mathbb{Z}+1,m\mathbb{Z}+2,...,m\mathbb{Z}+(m-1) \} $$

I was told that this was equivalent to $$ [0],[1],[2],...,[m-1] $$ where the braces $[...]$ indicate an equivalence class.

Is there a way I can immediately recognize that this is in fact the case? Is this comparable to a quotient space where this would imply that e.g. $m\mathbb{Z}+1\sim 1$ because these elements only differ by an element of the sub-group $(m\mathbb{Z},+)$, i.e., by the element $m\mathbb{Z}$

Or is my assumption wrong in the first place?

(Sorry for my inaccurate syntax...Im nowhere near to be a mathematician)

Answer 1

Is there a way I can immediately recognize that this is in fact the case?

Yes, by definition of $[a]_m$:

$$\begin{align} [a]_m&=\{a+M\mid M\in m\Bbb Z\} \\ &=\{ a+mx\mid x\in \Bbb Z\}, \end{align}$$ so $[a]_m=[b]_m$ iff $m\mid (a-b)$, leading to a complete system of residues modulo $m$.