Multiply and Add
Given any positive integer \(n\), let \(p(n)\) be the product of the non-zero digits of \(n\). If \(n\) has only one digit, then \(p(n)\) is equal to that digit.
\(Let: S = p(1) + p(2) + p(3) +\ldots + p(999)\).
What is the largest prime factor of \(S\)?