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?

$4$

$22$

$26$

$30$

$104$

Answer: C

Concepts:decimal prime factorization

Difficulty rating: 1800

Solution:

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.

2015 AMC 12A Problem 15

Answer: C

Difficulty rating: 1800

Solution:

Problem 15 in Other Years