Question 5
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.) Let $$
\begin{aligned}S&=p(1)+p(2)+p(3)\\&\quad+\cdots+p(999).\end{aligned}
$$ What is the largest prime factor of $S?$
\begin{aligned}S&=p(1)+p(2)+p(3)\\&\quad+\cdots+p(999).\end{aligned}
$$ What is the largest prime factor of $S?$