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?

$23$

$24$

$25$

$26$

$27$

Answer: D

Concepts:decimal exponent place value

Difficulty rating: 1280

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 .

2020 AMC 10B Problem 12

Answer: D

Difficulty rating: 1280

Solution:

Problem 12 in Other Years