Solutions

If n!  = n(n − 1)(αn + 1) with α integer, then after simplifying by n(n − 1), we conclude that (n − 2)!  ≡ 1 (mod n). Conversely, if this congruence holds, then the given condition is satisfied, with \( \alpha =\frac{\left(n-2\right)!-1}{n}. \)