Division with Quotient and Remainder Euclid

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 $b.$

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

Edit or Add Notation Axiomatic Method

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Bibliography (further reading)

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