Question 98
Define $\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when
$$
\sum_{\substack{2 \leq n \leq 50 \\ \operatorname{gcd}(n, 50)=1}} \phi^{!}(n)
$$
is divided by $50$.
$$
\sum_{\substack{2 \leq n \leq 50 \\ \operatorname{gcd}(n, 50)=1}} \phi^{!}(n)
$$
is divided by $50$.