Modulo Arithmetic/Examples/Solutions to x^2 = x Modulo 6

Example of Modulo Arithmetic
The equation:
 * $x^2 = x \pmod 6$

has solutions:

Proof
The Cayley table for multiplication modulo $6$ can be presented as:


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

\times_6 & 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}$

The squares $x^2$ of each $x$ can be found on the main diagonal, where each element of $\Z_6$ can be inspected.

Checking each of these, we have: