> [!NOTE] Lemma (Integers form Euclidean Domain) > The [[Absolute value function|absolute value function]] is [[Euclidean Domain|Euclidean]] on the [[Integers|ring of integers]] $\mathbb{Z}$ ###### Proof that Integers form Euclidean Domain \[MA268\] If $a\mid b$ with $a,b$ non-zero integers then $b=ac$ where $c$ is a non-zero integer. Thus $|b|=|a|\cdot |c|\geq |a|$ as $|c|\geq{1}$. Moreover, by [[Division with remainder for integers]], for all $a,b\in R$ with $a\neq 0$, $a=bq+r$ with $q,r\in \mathbb{Z}$ and $0\leq|r|\leq |b|$. # Application(s) # Reference(s)