Multiply and Add

Logic Level 4

Given any 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.

\(Let: S = p(1) + p(2) + p(3) +\ldots + p(999)\).

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

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