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.

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Divisibility definition
Congruence modulo n definition
Residue class definition

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Tag	`0QRV8FPK5M`
URI	`m2://0QRV8FPK5M`
Aliases	`congruence-class`
Kind	definition
Writing	W1 stated
MSC	11A07
Defines	Residue class
Uses	Congruence modulo n

Residue class

Definition

Remarks

Reading path

Where this leads

Further reading

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