ELMO SL 2019 N4

A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$ . Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$ .

Proposed by Carl Schildkraut and Holden Mui