What is the value of $43^{1234567890} \mod 22$?

Question

How can one find out the value of $43^{1234567890} \mod 22\;?$

Can I just say that because $123456890$ is an even number I can calculate $43^2 \mod 22$, which is $1$?

Answer 1

It turns out to be true, but the way you phrase it seems a bit wrong.

So, just to avoid any confusion, you can calculate $43^2 \pmod {22}$ but only because $43 \equiv -1 \pmod {22}$ and ${(-1)}^n = {(-1)}^2$ for every even n.

In other words, $42^{1234567890} \equiv 42^2 \pmod {22}$ would have been false

Answer 2

Yes that’s fine, it suffices use that

$$a \equiv b \mod m \implies a^n \equiv b^n \mod m$$

and in that case $43 \equiv -1 \mod {22}$.