Integer and Fifth Power have same Last Digit

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

Then $n^5$ has the same last digit as $n$ when both are expressed in conventional decimal notation.

Proof
From Fermat's Little Theorem: Corollary 1:
 * $n^5 \equiv n \pmod 5$

Suppose $n \equiv 1 \pmod 2$.

Then from Congruence of Powers:
 * $n^5 \equiv 1^5 \pmod 2$

and so:
 * $n^5 \equiv 1 \pmod 2$

Similarly, suppose $n \equiv 0 \pmod 2$.

Then from Congruence of Powers:
 * $n^5 \equiv 0^5 \pmod 2$

and so:
 * $n^5 \equiv 0 \pmod 2$

Hence:
 * $n^5 \equiv n \pmod 2$

So we have, by Chinese Remainder Theorem:


 * $n^5 \equiv n \pmod {2 \times 5}$

and the result follows.