The following proposition was already proven by Euclid about 300 B.C. and is therefore also called a Euclidian division.
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: 2019-07-28 14:08:26 | by: bookofproofs | references: 
 Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013