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Given a positive integer nnn, let p(n)p(n)p(n) be the product of the non-zero digits of nnn. (If nnn has one digit, then p(n)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). S=p(1)+p(2)+⋯+p(999).
What is the largest prime factor of SSS?
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