# 问题及解答

## 设 $n$ 是正整数, 证明 $n$ 和 $n^5$ 具有相同的个位数.

Posted by haifeng on 2021-11-24 16:01:48 last update 2021-11-24 16:01:48 | Edit | Answers (1)

Remark:

Prove that if $n$ is an integer, then $n$ and $n^5$ have the same units digit (first digit from the right).

1

Posted by haifeng on 2021-11-24 17:07:13

$n^5-n=n(n^4-1)=n(n-1)(n+1)(n^2+1).$

(1) 若 $n=5k+1$, 则 $n-1=5k$, 从而 $5|(n^5-n)$.

(2) 若 $n=5k+2$, 则 $n^2+1=(5k+2)^2+1=25k^2+20k+5$, 从而 $5|(n^5-n)$.

(3) 若 $n=5k+3$, 则 $n^2+1=(5k+3)^2+1=25k^2+30k+10$, 从而 $5|(n^5-n)$.

(4) 若 $n=5k+4$, 则 $n+1=5k+5$, 从而 $5|(n^5-n)$.