(1) (a) Find an inverse of 50 modulo 209.
(b) Let $p$ be an odd prime. Find the gcd
$( (p+1)!, (p-2)! + p^2 )$.
(c) Let $k$ be a positive integer and $q_k$ be the $k$ th prime (so that $q_1 = 2$, $q_2 = 3$, $q_3 = 5$, $q_4 = 7$, ...). Show
that $q_k$ can not divide
$\frac{q_{k+1}!}{q_k!}$
The notation ! above denotes the factorial.