# Multiplication modulo

Now here we are going to define another new type of multiplication, which is known as “multiplication modulo $p$.” It can be written as $a{ \times _p}b$, where $a$ and $b$ are any integers and $p$ is a fixed positive integer.

$a{ \times _p}b = r,\,for\,0 \leqslant r \leqslant p$

Here $r$ is the least non-negative remainder when $ab$, i.e. the ordinary product of $a$ and $b$, is divided by $p$. For example $4{ \times _7}2 = 1$, since $4 \times 2 = 8 = 1\left( 7 \right) + 1$.