Difference between revisions of "2015 AMC 12A Problems/Problem 15"
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What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal? | What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal? | ||
| − | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math> |
==Solution== | ==Solution== | ||
Revision as of 12:58, 18 March 2015
Problem
What is the minimum number of digits to the right of the decimal point needed to express the fraction 
as a decimal?
Solution
We can rewrite the fraction as 
. Since the last digit of the numerator is odd, a 
is added to the right if the numerator is divided by 
, and this will continuously happen because , itself, is odd. Indeed, this happens twenty-two times since we divide by twenty-two times, so we will need 
more digits. Hence, the answer is 
.
Alternate Solution
Note that 
is not a multiple of or , and therefore shares no factors with the original denominator. Multiple the numerator and denominator of the fraction by 
to give 
. This fraction will require 
divisions by ten to write as a decimal, and since the original fraction is less than 
all of the digits will be to the right of the decimal point. Answer: 
See Also
| 2015 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 14 |
Followed by Problem 16 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
