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 $$a b \left({\bmod\, m}\right)$$.

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.