2015 AMC 12A Problem 15

Question

What is the minimum number of digits to the right of the decimal point needed to express the fraction

\dfrac{123456789}{2^{26}\cdot 5^4}

as a decimal?

4

22

26

30

104

The numerator shares no factor of

2

or

5

with the denominator

Multiply the top and bottom to turn the denominator into a power of

10

The numerator and denominator share no common factors. To write the fraction as a decimal, rewrite it with a power of

10

in the denominator; the smallest that works is

10^{26}:

\dfrac{123456789}{2^{26}\cdot 5^4} = \dfrac{123456789\cdot 5^{22}}{10^{26}}.

Since the numerator

123456789\cdot 5^{22}

is not divisible by

10,

the decimal has exactly

26

places after the point.

Thus, the correct answer is C.

Problem 15 in other years