Definition#

Let \(n \geq 2\) be an integer. If \(a,b \in \mathbb{Z}\), then we say that \(a\) is congruent to \(b\) modulo \(n\), and write \(a \equiv b \: \text{(mod } n \text{)}\), if \(n \mid (a-b)\); that is, if \(a\) and \(b\) have the same remainder when divided by \(n\).

Example#

As \(8 \mid (53-21)\) (\(8 \mid 32\)), we have \(53 \equiv 21 \: \text{(mod 8)}\). Putting this another way, \(53\) and \(21\) both have a remainder of \(5\) when divided by \(8\). We reduce the remainder and write \(53 \equiv 5 \text{(mod 8)}\) and \(21 \equiv 5 \text{(mod 8)}\).

Source#

Gregory T. Lee. Abstract Algebra. 1st ed. Springer Cham, 2018. pp. 27.