Wilson factorial congruence modulo a triangular number ($(p-1)!\equiv p-1(\mathrm{mod}p(p-1)/2)$)

For an odd prime number $p$, put $m=(p-1)/2$. Both $(p-1)!$ and $p-1$ are divisible by $m$. By Wilson's theorem, they are also congruent modulo $p$, because each is $-1$ modulo $p$. The coprime integers $m,p$ therefore both divide their difference, giving the displayed integer congruence. This combines the prime-modulus factorial identity with an elementary divisibility constraint.