A number theory problem by Akshat Sharda
Given a positive integer \(n\), let \(P(n)\) be the product of the nonzero digits of \(n\).
If \(n\) has only one digit, then \(P(n)\) is equal to that digit.
\[S = P(1)+ P(2)+\ldots+P(999)\]
Find the largest prime factor of \(S\) ?