Difference between revisions of "2006 AIME I Problems/Problem 6"
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== Problem == | == Problem == | ||
| − | Let <math> \mathcal{S} </math> be the set of real | + | Let <math> \mathcal{S} </math> be the set of [[real number]]s that can be represented as repeating [[decimal notation| decimals]] of the form <math> 0.\overline{abc} </math> where <math> a, b, c </math> are distinct [[digit]]s. Find the sum of the elements of <math> \mathcal{S}. </math> |
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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 | + | 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> such numbers. 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>. |
Revision as of 09:53, 29 July 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 
such numbers. 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 
.
