2005 AIME I Problems/Problem 2

Problem

For each positive integer $k,$ let $S_k$ denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is $k.$ For example, $S_3$ is the sequence $1,4,7,10,\ldots.$ For how many values of $k$ does $S_k$ contain the term 2005?

Solution

Suppose that the nth term of the sequence $S_k$ is 2005. Then $1+(n-1)k=2005$ so $k(n-1)=2004=2^2\cdot 3\cdot 167.$ The ordered pairs $(k,n-1)$ of positive integers that satisfy the last equation are $(1,2004)$ , $(2,1002$ , $(3,668)$ , $(4,501)$ , $(6,334)$ , $(12,167)$ , $(167,12)$ , $(334,6)$ , $(501,4)$ , $(668,3)$ , $(1002,2)$ , and $(2004,1)$ . Thus the requested number of values is 12.

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2005 AIME I Problems/Problem 2

Problem

Solution

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