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.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-2}$ Fermat's Little Theorem: Exercise $3$