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

Example of Modulo Arithmetic

The equation:

$x^2 = x \pmod 6$

has solutions:

 $\displaystyle x$ $=$ $\displaystyle 0$ $\displaystyle x$ $=$ $\displaystyle 1$ $\displaystyle x$ $=$ $\displaystyle 3$ $\displaystyle x$ $=$ $\displaystyle 4$

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:

 $\displaystyle 0^2$ $=$ $\displaystyle 0$ $\displaystyle$ $\equiv$ $\displaystyle 0$ $\displaystyle \pmod 6$ $\displaystyle 1^2$ $=$ $\displaystyle 1$ $\displaystyle$ $\equiv$ $\displaystyle 1$ $\displaystyle \pmod 6$ $\displaystyle 2^2$ $=$ $\displaystyle 4$ $\displaystyle$ $\equiv$ $\displaystyle 4$ $\displaystyle \pmod 6$ $\displaystyle 3^2$ $=$ $\displaystyle 9$ $\displaystyle$ $\equiv$ $\displaystyle 3$ $\displaystyle \pmod 6$ $\displaystyle 4^2$ $=$ $\displaystyle 16$ $\displaystyle$ $\equiv$ $\displaystyle 4$ $\displaystyle \pmod 6$ $\displaystyle 5^2$ $=$ $\displaystyle 25$ $\displaystyle$ $\equiv$ $\displaystyle 1$ $\displaystyle \pmod 6$

$\blacksquare$