Definition:Modulo Multiplication

Definition
Let $m \in \Z$.

Let $\Z_m$ be the set of integers modulo $m$.

We define the multiplication operation on $\Z_m$ by the rule:


 * $\left[\!\left[{a}\right]\!\right]_m \times_m \left[\!\left[{b}\right]\!\right]_m = \left[\!\left[{a b}\right]\!\right]_m$

This is a well-defined operation.

This operation is called multiplication modulo $m$.

Comment
Although the operation of multiplication modulo $m$ is denoted by the symbol $\times_m$, if there is no danger of confusion, the conventional multiplication symbols $\times, \cdot$ etc. are often used instead.

In fact, the notation for multiplication of two integers modulo $m$ is not usually $\left[\!\left[{a}\right]\!\right]_m \times_m \left[\!\left[{b}\right]\!\right]_m$.

What is more normally seen is $a b \pmod m$.

Using this notation, what this result says is:

and it can be proved in the same way.

Warning
Note that while the modulo operation is defined for all real numbers, the operation of modulo multiplication $\times_m$ is defined only when $a, b, m$ are all integers.

The reason for this can be found here.

Also see

 * Modulo Multiplication is Well-Defined