The integers are a Euclidean domain

Statement

The ring of integers $\mathbb{Z}$ is a Euclidean domain, with the absolute value $\delta(n) = |n|$ as Euclidean function. That is, for all $a, b \in \mathbb{Z}$ with $b \neq 0$, there are $q, r \in \mathbb{Z}$ with $a = b q + r$ and $0 \le r < |b|$.

Remarks

The proof: let $r$ be the least element of $\{a - b q : q \in \mathbb{Z}\} \cap \mathbb{N}$. Then $r < |b|$, because otherwise $r - |b|$ would be a smaller element of this set. The pair $(q, r)$ with $0 \le r < |b|$ is unique. Consequences: $\mathbb{Z}$ is a principal ideal domain, and integers factor uniquely into primes.

Explanations

L1 human

This is division with remainder from school, made precise. Dividing $17$ by $5$ gives $17 = 3 \cdot 5 + 2$, and the remainder $2$ is smaller than $5$. It works for negative numbers too: $-17 = (-4) \cdot 5 + 3$.