2002 AIME I Problem 14

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?