Integer and Fifth Power have same Last Digit
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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 $3$