A problem I have been presented with asks the following:
Prove for every odd number $x$, $ x^2$ is always congruent to $1$ or $9$ modulo $24$.
This seems odd and non-intuitive to me. Of course, it must be true other wise they wouldn't be asking for me to prove it.
I know that:
$9$ modulo $24$ $=$ $9$
$1 = 1$
How could every odd number in existence squared be equal to either 1 or 9?