2020 AMC 10B 第 12 题

Question

数

\frac{1}{20^{20}}

的小数表示在小数点后先出现一串零，然后是一个

9

，后面还有若干数字。小数点后最初的一串零有多少个？

The decimal representation of

\frac{1}{20^{20}}

consists of a string of zeros after the decimal point, followed by a

9

and then several more digits. How many zeros are in that initial string of zeros after the decimal point?

23

24

25

26

27

把

\frac{1}{20^{20}}

改写成

\frac{5^{20}}{10^{40}}

。

Rewrite

\frac{1}{20^{20}}

as

\frac{5^{20}}{10^{40}}

求

5^{20}

有多少位，就能知道前导零的个数。

Find how many digits

5^{20}

has to know how many leading zeros appear

可以写成

\frac{1}{20^{20}}=\frac{1}{(4\cdot5)^{20}}=\frac{5^{20}}{10^{40}}\text{。}

又因为

5^{20}=95{,}367{,}431{,}640{,}625

，这是一个

14

位数，且首位为

9

。除以

10^{40}

后，将这个

14

位数放在小数点后，并在首位数字前放置

40-14=26

个零。

所以正确答案是 D。

We can write

\frac{1}{20^{20}}=\frac{1}{(4\cdot5)^{20}}=\frac{5^{20}}{10^{40}}.

Now

5^{20}=95{,}367{,}431{,}640{,}625,

which has

14

digits and begins with

9.

Dividing by

10^{40}

places this

14

-digit number after the decimal point with

40-14=26

zeros before the first digit.

Thus, the correct answer is D .

其他年份的第 12 题