Integers modulo n as a quotient group

Example

Let $n$ be a positive integer. The subgroup $n\mathbb{Z}$ of the additive group of integers is normal, because $\mathbb{Z}$ is abelian. The quotient group $\mathbb{Z}/n\mathbb{Z}$ has the $n$ cosets $0 + n\mathbb{Z}, 1 + n\mathbb{Z}, \ldots, (n-1) + n\mathbb{Z}$. The operation is $(a + n\mathbb{Z}) + (b + n\mathbb{Z}) = (a + b) + n\mathbb{Z}$, which is addition modulo $n$.

Remarks

The coset $a + n\mathbb{Z}$ is the set of integers with the same remainder as $a$ on division by $n$. Thus $\mathbb{Z}/n\mathbb{Z}$ is the cyclic group of order $n$ of clock arithmetic. For $n = 0$, the quotient $\mathbb{Z}/0\mathbb{Z}$ is isomorphic to $\mathbb{Z}$.

Explanations

L1 human

Two integers are in the same coset of $n\mathbb{Z}$ when their difference is a multiple of $n$. So the quotient group keeps only the remainder modulo $n$. Adding cosets is the same as adding remainders and reducing modulo $n$.