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?
(第二十三届AIME1 2005 第2题)