Modulo Multiplication/Cayley Table/Modulo 5

Cayley Table for Modulo Multiplication

The multiplicative monoid of integers modulo $m$ can be described by showing its Cayley table.

This one is for modulo $5$:

$\begin{array} {r|rrrrr} \struct {\Z_5, \times_5} & \eqclass 0 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \hline \eqclass 0 5 & \eqclass 0 5 & \eqclass 0 5 & \eqclass 0 5 & \eqclass 0 5 & \eqclass 0 5 \\ \eqclass 1 5 & \eqclass 0 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \eqclass 2 5 & \eqclass 0 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 1 5 & \eqclass 3 5 \\ \eqclass 3 5 & \eqclass 0 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 4 5 & \eqclass 2 5 \\ \eqclass 4 5 & \eqclass 0 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 2 5 & \eqclass 1 5 \\ \end{array}$

which can also be presented:

$\begin{array} {r|rrrrr} \times_5 & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 \\ 2 & 0 & 2 & 4 & 1 & 3 \\ 3 & 0 & 3 & 1 & 4 & 2 \\ 4 & 0 & 4 & 3 & 2 & 1 \\ \end{array}$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 19$: Properties of $\Z_m$ as an algebraic system
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): modulo $n$, addition and multiplication