Modulo Multiplication/Cayley Table

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 $6$:


 * $\begin{array}{r|rrrrrr}

\left({\Z_6, \times_6}\right) & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 \\ \hline \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 \\ \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 \\ \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 \\ \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 \\ \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 \\ \left[\!\left[{5}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 \\ \end{array}$

which can also be presented:


 * $\begin{array}{r|rrrrrr}

\cdot_m & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 \\ 2 & 0 & 2 & 4 & 0 & 2 & 4 \\ 3 & 0 & 3 & 0 & 3 & 0 & 3 \\ 4 & 0 & 4 & 2 & 0 & 4 & 2 \\ 5 & 0 & 5 & 4 & 3 & 2 & 1 \\ \end{array}$