Modulo Multiplication is Commutative

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Theorem

Multiplication modulo $m$ is commutative:

$\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m = \eqclass y m \times_m \eqclass x m$


Proof

\(\displaystyle \eqclass x m \times_m \eqclass y m\) \(=\) \(\displaystyle \eqclass {x y} m\) Definition of Modulo Multiplication
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {y x} m\) Integer Multiplication is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \eqclass y m \times_m \eqclass x m\) Definition of Modulo Multiplication

$\blacksquare$


Sources