jmc

Question

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,. For how many values of k does S_k contain 2005 as a term?

Accepted Answer

The general term of the sequence is a_n = 1 + kn, where a_0 = 1 is the first term. Therefore, we want 1 + kn = 2005, or kn = 2004. We see that this equation has a solution for n if and only if k is a divisor of 2004. Since 2004 = 2^2 3 167, the number of positive divisors of 2004 is (2+1)(1+1)(1+1) = 12.