1994 AIME Problem 5

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\,?

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.