IMO training problem

Logic Level 3

Given a positive integer nn, let p(n)p(n) be the product of the non-zero digits of nn. (If nn has one digit, then p(n)p(n) is equal to that digit.) Let

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

What is the largest prime factor of SS?

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