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 $S$ ?