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?

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?