# Modulo Multiplication has Identity

## Theorem

$\forall \eqclass x m \in \Z_m: \eqclass x m \times_m \eqclass 1 m = \eqclass x m = \eqclass 1 m \times_m \eqclass x m$

## Proof

Follows directly from the definition of multiplication modulo $m$:

 $\ds \eqclass x m \times_m \eqclass 1 m$ $=$ $\ds \eqclass {x \times 1} m$ $\ds$ $=$ $\ds \eqclass x m$ $\ds$ $=$ $\ds \eqclass {1 \times x} m$ $\ds$ $=$ $\ds \eqclass 1 m \times_m \eqclass x m$

Thus $\eqclass 1 m$ is the identity for multiplication modulo $m$.

$\blacksquare$