Division algorithm

Statement

Let $a$ and $b$ be integers with $b \neq 0$. Then there is exactly one pair of integers $(q, r)$ such that $a = b q + r$ and $0 \le r < |b|$.

Remarks

The integer $q$ is the quotient and the integer $r$ is the remainder of the division of $a$ by $b$. The remainder is $0$ if and only if $b$ divides $a$. For example, $-7 = 3 \cdot (-3) + 2$, so the quotient of $-7$ by $3$ is $-3$ and the remainder is $2$.

Explanations

L1 human

You can share $a$ objects into groups of $b$ objects, with some objects left over. The theorem says that the number of full groups and the rest are unique, if the rest is smaller than $|b|$. For example, $17 = 5 \cdot 3 + 2$: you get $3$ groups of $5$, and $2$ objects stay.