Clock Squares

If you have already met More Adventures with Modular Arithmetic then you will have learnt that if $A \equiv a \text{ mod } n$ and $B \equiv b \text{ mod } n$ then $AB \equiv ab \text{ mod } n$. 

This means that since $23 \equiv 2 \text{ mod } 7$ we have $23^2 \equiv 2^2 \text{ mod }7$ and so $23^2 \equiv 4 \text{ mod }7$.