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)