2015 AMC 12A Problems/Problem 15

Problem

What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{123456789}{2^{26}\cdot 5^4}$ as a decimal?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104$

Solution

We can rewrite the fraction as $\frac{123456789}{2^{22} \cdot 10^4} = \frac{12345.6789}{2^{22}}$ . Since the last digit of the numerator is odd, a $5$ is added to the right if the numerator is divided by $2$ , and this will continuously happen because $5$ , itself, is odd. Indeed, this happens twenty-two times since we divide by $2$ twenty-two times, so we will need $22$ more digits. Hence, the answer is $4 + 22 = 26 \textbf{ (C)}$ .

Alternate Solution

Note that $123456789$ is not a multiple of $2$ or $5$ , and therefore shares no factors with the original denominator. Multiple the numerator and denominator of the fraction by $5^{22}$ to give $\frac{5^{22} \cdot 123456789}{10^{26}}$ . This fraction will require $26$ divisions by ten to write as a decimal, and since the original fraction is less than $1$ all of the digits will be to the right of the decimal point. Answer: $\textbf{ (C)}$

2015 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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2015 AMC 12A Problems/Problem 15

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Solution

Alternate Solution

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