Finding solutions of $x^{201}+x^{21}+x+1 \equiv 0 \mod 5$?

Question

Find integral solutions of:

$x^{201}+x^{21}+x+1 \equiv 0 \mod 5$.

I know the solution is $3+5n, n \in \mathbb Z $ but I'm trying to work out how to get there.

The big exponents are throwing me off, what is the easiest method? I thought about plugging in the values of $\mod 5$ manually...

Answer 1

HINT

Note that by FLT

$$x^4\equiv 1 \mod 5$$

then

$$x^{201}+x^{21}+x+1\equiv x+x+x+1 \equiv 3x+1 \mod 5$$