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Difference between revisions of "2016 AMC 12B Problems/Problem 22"

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=Solution=
 
=Solution=
 
Solution by e_power_pi_times_i
 
Solution by e_power_pi_times_i
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If <math>\frac{1}{n} = abcdef</math> and <math>n < 1000</math>, <math>n</math> must be a factor of <math>999999</math>. Also, by the same procedure, <math>n+6</math> must be a factor of <math>9999</math>. Checking through all the factors of <math>999999</math> and <math>9999</math>, we see that <math>n =297</math> is a solution, so the answer is .
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If <math>\frac{1}{n} = abcdef</math> and <math>n < 1000</math>, <math>n</math> must be a factor of <math>999999</math>. Also, by the same procedure, <math>n+6</math> must be a factor of <math>9999</math>. Checking through all the factors of <math>999999</math> and <math>9999</math>, we see that <math>n =297</math> is a solution, so the answer is <math>\boxed{(B)}</math>.
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2016|ab=B|num-b=21|num-a=23}}
 
{{AMC12 box|year=2016|ab=B|num-b=21|num-a=23}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 00:28, 22 February 2016

Problem

For a certain positive integer n less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period of 6, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period 4. In which interval does $n$ lie?

$\textbf{(A)}\ [1,200]\qquad\textbf{(B)}\ [201,400]\qquad\textbf{(C)}\ [401,600]\qquad\textbf{(D)}\ [601,800]\qquad\textbf{(E)}\ [801,999]$

Solution

Solution by e_power_pi_times_i If $\frac{1}{n} = abcdef$ and $n < 1000$, $n$ must be a factor of $999999$. Also, by the same procedure, $n+6$ must be a factor of $9999$. Checking through all the factors of $999999$ and $9999$, we see that $n =297$ is a solution, so the answer is $\boxed{(B)}$.

See Also

2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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

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