The following proposition was already proven by Euclid about 300 B.C. and is therefore also called a **Euclidian** division.

**Lemma**: Division with Quotient and Remainder (Euclid)

Let $a,b\in\mathbb Z$ be integers. If $a > 0$ then there are uniquely determined integers $q,r$ with $$b=qa+r,\quad 0\le r< a.$$

We call the number $r$ a **remainder**, the number $q$ is called the **quotient** of the division. In the special case $r=0$ we have that $a\mid b,$ i.e. $a$ is a divisor of $b.$

| | | | | created: 2014-08-30 09:44:26 | modified: 2019-03-09 19:27:34 | by: *bookofproofs* | references: [696]

## 1.**Proof**: *(related to "Division with Quotient and Remainder")*

(none)

[696] **Kramer Jürg, von Pippich, Anna-Maria**: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013

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