Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.
Problem
Source: 2021 Saudi Arabia Training Lists p39 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests
Tags: Sum, number theory, divides, divisible