Question

$1999^{2000}$ 除以 $5$ 的余数是

When $1999^{2000}$ is divided by $5,$ the remainder is

$4$

$3$

$2$

$1$

$0$

答案：D

知识点：模运算个位数字

难度评级：1470

解答：

要求除以 $5$ 的余数，只需要关注个位数。

因此只需观察个位数 $9$ 的幂的规律。

$9$ 的幂的个位数在 $9$ 和 $1$ 之间交替：因为指数是偶数，所以 $1999^{2000}$ 的个位数为 $1$ 。 $9, 81, 729, \cdots$

除以 $5$ 的余数就是 $1$ ，因为它比 $10$ 的倍数多 $1$ 。

所以正确答案是 D。

Note that to find the remainder when divided by $5,$ we only care about the units digit.

This means we only have to observe how the powers of the units digit work, namely the powers of $9.$

Then, looking at powers of $9,$ we see that the units digit alternates between $9$ and $1:$ $9, 81, 729, \cdots$ This means that $1999^{2000}$ ends in a $1$ since the power is even.

The remainder when divided by $5$ is then $1,$ since it is $1$ more than a multiple of $10.$

Thus, D is the correct answer.

1999 AMC 8 第 24 题

答案：D

难度评级：1470

解答：

其他年份的第 24 题