# Definitions > [!NOTE] Definition (Congruence Modulo $n$) >Take [[Integers|integers]] $a,b,n\in\mathbb{Z}.$ We say that $a$ is *congruent modulo $n$* to $b$, and write $a\equiv b \mod{n}$if $n\mid(a-b).$ > > [!NOTE] Lemma ($a$ and $b$ are congruent iff they have the same remainder after division by $n)$ > -- [[Division with remainder for integers]] *Proof*. ... # Properties > [!NOTE] Lemma (Congruence is an equivalence relation) >Congruence modulo $n$ is an [[Equivalence relations|equivalence relation]] on $\mathbb{Z}.$ *Proof*. Since $n|0,$ congruence is reflexive. If $n|(x-y)$ then $x-y = an \iff y-x=-an$ for some integer $a.$ So $n\mid(y-x)$ and congruence is symmetric. Lastly suppose $n\mid(x-y)$ and $n\mid(y-z),$ then $x-y=an$ and $y-z=bn$ for some integers $a$ and $b.$ Adding both equations gives $x-z=(a+b)n$ and so $n\mid(x-z)$ which shows that congruence is transitive. The equivalence class of an integer is known as its [[Congruence Class|congruence class]] modulo $n.$ The quotient set of $\mathbb{Z}$ by congruence modulo $n$ is called '[[Integers modulo n|Z modulo n Z]]', denoted $\mathbb{Z}/n\mathbb{Z}.$ # Applications **Linear congruences**: A [[Solutions to linear congruence|linear congruence]] is an equation of the form $ax \equiv b \pmod{n}.$ The [[Chinese remainder theorem|Chinese remainder theorem]] asserts that there is a unique solution to any system of linear congruences provided the moduli are pairwise coprime. **Generalisations**: [[Congruence Modulo Subgroup]].