A number theory problem by Akshat Sharda

Given a positive integer \(n\), let \(P(n)\) be the product of the nonzero digits of \(n\).

If \(n\) has only one digit, then \(P(n)\) is equal to that digit.

\[S = P(1)+ P(2)+\ldots+P(999)\]

Find the largest prime factor of \(S\) ?

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