About

If two numbers $b$ and $c$ have the property that their difference $b-c$ is integrally divisible by a number $m$ (i.e., $(b-c)/m$ is an integer), then $b$ and $c$ are said to be "congruent modulo $m$."

The number $m$ is called the modulus, and the statement "$b$ is congruent to $c$ (modulo $m$)"; mathematically written as $b≡c \pmod{m}$.

If $b-c$ is not integrally divisible by $m$, then it is said that "$b$ is not congruent to $c$ (modulo $m$),"; mathematically written as $b≢c \pmod{m}$.

The explicit $\pmod{m}$ is sometimes omitted when the modulus $m$ is understood by context, so in such cases, care must be taken not to confuse the symbol $≡$ with the equivalence sign.

The quantity $b$ is sometimes called the "base," and the quantity $c$ is called the residue or remainder. There are several types of residues. The common residue defined to be non-negative and smaller than $m$, while the minimal residue is $c$ or $c-m$, whichever is smaller in absolute value.

history | show excerpt | excerpt history