2020 AMC 10B Problem 12

Question

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

Rewrite

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

as

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

Find how many digits

5^{20}

has to know how many leading zeros appear

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 .

Problem 12 in other years