# Square of Modulo less One equals One

## Theorem

Let $m \in \Z$ be an integer.

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

$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

Then:

$\eqclass {m - 1} m \times_m \eqclass {m - 1} m = \eqclass 1 m$

where $\times_m$ denotes multiplication modulo $m$.

## Proof

 $\ds \eqclass {m - 1} m \times_m \eqclass {m - 1} m$ $=$ $\ds \eqclass {\paren {m - 1}^2} m$ Definition of Modulo Multiplication $\ds$ $=$ $\ds \eqclass {m^2 - 2 m + 1} m$ $\ds$ $=$ $\ds \eqclass 1 m$

$\blacksquare$