Definition:Modulo Multiplication

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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:

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle a\) \(\equiv\) \(\displaystyle \) \(\displaystyle b\) \(\displaystyle \pmod m\) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle c\) \(\equiv\) \(\displaystyle \) \(\displaystyle d\) \(\displaystyle \pmod m\) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \implies\) \(\displaystyle \) \(\displaystyle a c\) \(\equiv\) \(\displaystyle \) \(\displaystyle b d\) \(\displaystyle \pmod m\) \(\displaystyle \)                    

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


Sources

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