# Difference between revisions of "2005 AIME I Problems/Problem 2"

m |
|||

Line 1: | Line 1: | ||

== Problem == | == Problem == | ||

− | For each positive integer <math> k | + | For each [[positive integer]] <math>k</math>, let <math> S_k </math> denote the [[increasing sequence | increasing]] [[arithmetic sequence]] of [[integer]]s whose first term is 1 and whose common difference is <math> k</math>. For example, <math> S_3 </math> is the [[sequence]] <math> 1,4,7,10,\ldots. </math> For how many values of <math> k </math> does <math> S_k </math> contain the term 2005? |

== Solution == | == Solution == | ||

− | Suppose that the | + | Suppose that the <math>n</math>th term of the sequence <math>S_k</math> is 2005. Then <math>1+(n-1)k=2005</math> so <math>k(n-1)=2004=2^2\cdot 3\cdot 167</math>. The [[ordered pair]]s <math>(k,n-1)</math> of positive integers that satisfy the last equation are <math>(1,2004)</math>,<math>(2,1002)</math>, <math>(3,668)</math>, <math>(4,501)</math>, <math>(6,334)</math>, <math>(12,167)</math>, <math>(167,12)</math>,<math>(334,6)</math>, <math>(501,4)</math>, <math>(668,3)</math>, <math>(1002,2)</math> and <math>(2004,1)</math>, and each of these gives a possible value of <math>k</math>. Thus the requested number of values is 12. |

== See also == | == See also == | ||

+ | * [[2005 AIME I Problems/Problem 1 | Previous problem]] | ||

+ | * [[2005 AIME I Problems/Problem 3 | Next problem]] | ||

* [[2005 AIME I Problems]] | * [[2005 AIME I Problems]] | ||

+ | |||

+ | [[Category:Introductory Number Theory Problems]] | ||

+ | [[Category:Introductory Algebra Problems]] |

## Revision as of 12:48, 29 November 2006

## Problem

For each positive integer , let denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is . For example, is the sequence For how many values of does contain the term 2005?

## Solution

Suppose that the th term of the sequence is 2005. Then so . The ordered pairs of positive integers that satisfy the last equation are ,, , , , , ,, , , and , and each of these gives a possible value of . Thus the requested number of values is 12.