Given a 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

$$S=p(1)+p(2)+p(3)+\cdots+p(999)$$.

What is the largest prime factor of \(S\,\)?

(第十二届AIME1994 第5题)