> [!NOTE] **Definition** (Unit of $\mathbb{Z}/n\mathbb{Z}$) > A congruence class $[a]_{m} \in \mathbb{Z} \text{/} m\mathbb{Z}$ ([[Integers modulo n|Z mod m Z]]) is a *[[Unit in a Ring|unit]]* if it has a multiplicative inverse in $\mathbb{Z} \text{/}m\mathbb{Z}$. # Properties **Algebra**: the [[Unit Group of Integers Modulo n|set of units]] of $\mathbb{Z}/n\mathbb{Z},$ denoted $(\mathbb{Z}/n\mathbb{Z})^{\times},$ is a multiplicative group known as the unit group of $\mathbb{Z}/n\mathbb{Z}.$ **Tests for unit**: By [[Unit modulo n iff coprime to n|condition for unit modulo n]], $[a]_{n}$ is a unit off $\mathbb{Z}/n\mathbb{Z}$ iff $\gcd(a,n)=1.$ **Order of a unit**: By [[Euler's theorem (Number Theory)|Euler's theorem]], the order of a unit of $\mathbb{Z}/n\mathbb{Z}$ divides $\varphi(n),$ the order of the unit group ($\varphi$ denotes [[Euler Totient Function|Euler totient function]]). In the special case that $n$ is prime, Euler's theorem is known as [[Fermat's little theorem]]. # Applications **Generalisations**: (1) Note that in general, the order of any member of a finite group divides the order of the group by [[Order of Element of Finite Group Divides Order of The Group|Lagrange's theorem]].