Question

$(20!-15!)$ 的百位数字是多少？

What is the hundreds digit of $(20!-15!)?$

$0$

$1$

$2$

$4$

$5$

只需要考虑它是否能被 $1000$ 整除。

Look only at divisibility by $1000$

两个阶乘都至少含有三个因子 $5$ 和三个因子 $2$ 。

Both factorials contain at least three factors of $5$ and three factors of $2$

$20!$ 和 $15!$ 都至少含有三个因数 $5$ 和三个因数 $2$ ，所以两者都能被 $2^3\cdot5^3=1000$ 整除。

因此它们的差也能被 $1000$ 整除。

一个数是 $1000$ 的倍数，说明末三位都是 $0$ ，所以百位数字也是 $0$ 。

所以正确答案是 A。

Both $20!$ and $15!$ contain at least three factors of $5$ and three factors of $2$ , so both are divisible by $2^3\cdot5^3=1000.$

Their difference is therefore also divisible by $1000.$

Being a multiple of $1000$ makes the last three digits $0,$ which shows that the hundreds digit is also $0.$

Thus, A is the correct answer.