# 2005 AIME I Problems/Problem 2

## 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.