Residue class

Definition

Let $n$ be a positive integer and let $a$ be an integer. The residue class of $a$ modulo $n$ is the set $\overline{a} = \{ b \in \mathbb{Z} : b \equiv a \pmod n \}$ of the integers congruent to $a$ modulo $n$. Thus $\overline{a} = \{ a + k n : k \in \mathbb{Z} \} = a + n \mathbb{Z}$.

Remarks

Two residue classes $\overline{a}$ and $\overline{b}$ are equal if $a \equiv b \pmod n$, and they have no common element otherwise. There are exactly $n$ residue classes modulo $n$: $\overline{0}, \overline{1}, \dots, \overline{n - 1}$. Other notations are $[a]$, $[a]_n$ and $a \bmod n$. The residue classes are the elements of the ring of integers modulo n.

Explanations

L1 human

Modulo $3$, each integer falls in one of three boxes: the multiples of $3$, the numbers $1$ more than a multiple of $3$, and the numbers $2$ more. Each box is a residue class. For example, $\overline{1} = \{ \dots, -5, -2, 1, 4, 7, \dots \}$.