Difference between revisions of "2006 AIME I Problems/Problem 6"
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== Solution == | == Solution == | ||
| − | + | Numbers of the form <math>0.\overline{abc}</math> can be written as <math>\frac{abc}{999}</math>. There are <math>10\times9\times8=720</math> numbers total. Each digit will appear in each place value <math>\frac{720}{10}=72</math> times, and the sum of the digits, 0 through 9, is 45. So the sum of all the numbers is <math>\frac{45\times72\times111}{999}=360</math>. | |
== See also == | == See also == | ||
* [[2006 AIME I Problems]] | * [[2006 AIME I Problems]] | ||
Revision as of 23:41, 30 June 2006
Problem
Let 
be the set of real numbers that can be represented as repeating decimals of the form 
where 
are distinct digits. Find the sum of the elements of 
Solution
Numbers of the form can be written as 
. There are 
numbers total. Each digit will appear in each place value 
times, and the sum of the digits, 0 through 9, is 45. So the sum of all the numbers is 
.
