2024 USAMO Problems/Problem 2

Let $S_1, S_2, \ldots, S_{100}$ be finite sets of integers whose intersection is not empty. For each non-empty $T \subseteq\left\{S_1, S_2, \ldots, S_{100}\right\}$ , the size of the intersection of the sets in $T$ is a multiple of the number of sets in $T$ . What is the least possible number of elements that are in at least 50 sets?