Modulo operation

Definition

Let $a$ and $n$ be integers with $n \neq 0$. The integer $a \bmod n$ is the remainder $r$ of the division of $a$ by $n$. Thus $a \bmod n$ is the unique integer $r$ such that $0 \le r < |n|$ and $a - r$ is a multiple of $n$.

Remarks

For example, $17 \bmod 5 = 2$ and $(-17) \bmod 5 = 3$. Many programming languages use a different convention. In them, the remainder has the sign of $a$, so the result for $-17$ and $5$ is $-2$. The Lean library Mathlib sets $a \bmod 0 = a$.

Explanations

L1 human

The value $a \bmod n$ is what is left after you take away from $a$ as many copies of $n$ as you can. A clock uses this operation: $15$ hours after $0$ o'clock, a $12$-hour clock shows $15 \bmod 12 = 3$.