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 $b.$
Proofs: 1
Definitions: 1 2
Proofs: 3 4 5 6 7 8 9 10 11 12
Propositions: 13 14 15 16 17 18
Solutions: 19 20 21