Addition modulo an integer ($[a{]}_n+[b{]}_n=[a+b{]}_n$)

For an integer $n\ge 1$, add residue classes by adding their representatives and taking the class modulo $n$. This is well-defined because replacing either representative by a multiple-of-$n$ translate changes the sum by a multiple of $n$. The associativity, identity class $[0{]}_n$ and inverse $[-a{]}_n$ are inherited from integer addition. The result is the additive cyclic group $\mathbb{Z}/n\mathbb{Z}$.