2020 AMC 10B Problem 12

Problem 12 of 25IntermediateAlgebra Number Theory

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?

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Solution

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 .