Congruence modulo n

Definition

Let $n$ be a positive integer, and let $a$ and $b$ be integers. The integer $a$ is congruent to $b$ modulo $n$ when $n$ divides $a - b$. The notation is $a \equiv b \pmod{n}$. The integer $n$ is the modulus of the congruence.

Remarks

Equivalently, $a \equiv b \pmod n$ if and only if $a$ and $b$ have the same remainder on division by $n$, that is, $a \bmod n = b \bmod n$. Congruence modulo $n$ is an equivalence relation on the integers. Its classes are the residue classes modulo $n$. For example, $38 \equiv 14 \pmod{12}$, because $12 \mid 24$. Some texts allow each integer $n$ as a modulus. Then congruence modulo $0$ is equality, and congruence modulo $-n$ is congruence modulo $n$.

Explanations

L1 human

Two integers are congruent modulo $n$ when they differ by a multiple of $n$. On a $12$-hour clock, $14$ o'clock and $2$ o'clock show the same time, because $14 \equiv 2 \pmod{12}$.