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Difference between revisions of "2005 AIME I Problems/Problem 12"

 
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== Problem ==
 
== Problem ==
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For positive integers <math> n, </math> let <math> \tau (n) </math> denote the number of positive integer divisors of <math> n, </math> including 1 and <math> n. </math> For example, <math> \tau (1)=1 </math> and <math> \tau(6) =4. </math> Define <math> S(n) </math> by <math> S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n). </math> Let <math> a </math> denote the number of positive integers <math> n \leq 2005 </math> with <math> S(n) </math> odd, and let <math> b </math> denote the number of positive integers <math> n \leq 2005 </math> with <math> S(n) </math> even. Find <math> |a-b|. </math>
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For [[positive integer]]s <math> n, </math> let <math> \tau (n) </math> denote the number of positive integer [[divisor]]s of <math> n, </math> including 1 and <math> n. </math> For example, <math> \tau (1)=1 </math> and <math> \tau(6) =4. </math> Define <math> S(n) </math> by <math> S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n). </math> Let <math> a </math> denote the number of positive integers <math> n \leq 2005 </math> with <math> S(n) </math> [[odd integer | odd]], and let <math> b </math> denote the number of positive integers <math> n \leq 2005 </math> with <math> S(n) </math> [[even integer | even]]. Find <math> |a-b|. </math>
  
 
== Solution ==
 
== Solution ==
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{{solution}}
  
 
== See also ==
 
== See also ==
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* [[2005 AIME I Problems/Problem 13 | Next problem]]
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* [[2005 AIME I Problems/Problem 11 | Previous problem]]
 
* [[2005 AIME I Problems]]
 
* [[2005 AIME I Problems]]
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[[Category:Intermediate Number Theory Problems]]

Revision as of 16:21, 12 October 2006

Problem

For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\tau (1)=1$ and $\tau(6) =4.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 2005$ with $S(n)$ even. Find $|a-b|.$

Solution

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See also

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