# Multiplicative Group of Reduced Residues/Examples/Modulo 8

## Example of Multiplicative Group of Reduced Residues

Consider the reduced residue system $\Z'_8$ modulo $8$ under modulo multiplication:

$\Z'_8 = \set {\eqclass 1 8, \eqclass 3 8, \eqclass 5 8, \eqclass 7 8}$

$\struct {\Z'_8, \times_8}$ is the multiplicative group of reduced residues modulo 8.

### Cayley Table

$\begin{array}{r|rrrr} \times_8 & \eqclass 1 8 & \eqclass 3 8 & \eqclass 5 8 & \eqclass 7 8 \\ \hline \eqclass 1 8 & \eqclass 1 8 & \eqclass 3 8 & \eqclass 5 8 & \eqclass 7 8 \\ \eqclass 3 8 & \eqclass 3 8 & \eqclass 1 8 & \eqclass 7 8 & \eqclass 5 8 \\ \eqclass 5 8 & \eqclass 5 8 & \eqclass 7 8 & \eqclass 1 8 & \eqclass 3 8 \\ \eqclass 7 8 & \eqclass 7 8 & \eqclass 5 8 & \eqclass 3 8 & \eqclass 1 8 \\ \end{array}$