A set \(\mathcal{S}\) of distinct positive integers has the following property: for every integer \(x\) in \(\mathcal{S},\) the arithmetic mean of the set of values obtained by deleting \(x\) from \(\mathcal{S}\) is an integer. Given that 1 belongs to \(\mathcal{S}\) and that 2002 is the largest element of \(\mathcal{S},\) what is the greatest number of elements that \(\mathcal{S}\) can have?
(第二十届AIME1 2002 第14题)