Fourth Power Modulo 5

Theorem
Let $n \in \Z$ be an integer.

Then:
 * $n^4 \equiv m \pmod 5$

where $m \in \set {0, 1}$.

Proof
By Congruence of Powers:
 * $a \equiv b \pmod 5 \iff a^4 \equiv b^4 \pmod 5$

so it is sufficient to demonstrate the result for $n \in \set {0, 1, 2, 3, 4}$.

Thus:

The result follows.