For every $n\in\mathbb N$, the congruence-class group $(\mathbb Z/n,+)$ is the quotient group $(\mathbb Z,+)/n\mathbb Z$

Given: A natural number $n$, viewed in $\mathbb Z$ under the canonical embedding, and the set $n\mathbb Z=\{nk:k\in\mathbb Z\}$.

The integers form a commutative ring with identity (The integers form a commutative ring), and the canonical embedding of $\mathbb N$ preserves addition and multiplication (The naturals embed in the integers).

A subset of a group is a subgroup when it contains the identity and is closed under the operation and inverses (Subgroup).

Every subgroup of an abelian group is normal (Every subgroup of an abelian group is normal).

The congruence $x\equiv a\pmod n$ means that $x-a=nq$ for some $q\in\mathbb Z$ (Congruence modulo an integer: $a\equiv b\pmod n$ when $n\mid(a-b)$, including the moduli $0$ and $1$).

The congruence class is $[a]_n=\{x\in\mathbb Z:x\equiv a\pmod n\}$ (The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$).

Addition modulo $n$ is $[a]_n+[b]_n=[a+b]_n$ (Addition and multiplication on $\mathbb{Z}/n$ by $[a]_n+[b]_n=[a+b]_n$ and $[a]_n[b]_n=[ab]_n$).

The cosets of a normal subgroup form a group under $(a+N)+(b+N)=(a+b)+N$ (For $N\mathrel{\trianglelefteq}G$, the cosets form a group with identity $N$ and inverse $(gN)^{-1}=g^{-1}N$).

For every $n\in\mathbb N$, $(\mathbb Z/n,+,[0]_n)$ is an abelian group, including at $n=0$ and $n=1$ (For every natural $n$, $(\mathbb{Z}/n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).