Wilson's Theorem (Gauss's Generalization)

Let $P$ be the product of Integers less than or equal to $n$ and Relatively Prime to $n$. Then

$$P\equiv \prod_{\scriptstyle k=2\atop\scriptstyle k\notdiv n}... ..., p^\alpha, 2p^\alpha$\cr 1\ ({\rm mod\ } n) & otherwise.\cr}$$

When $m=2$, this reduces to $P\equiv 1\ \left({{\rm mod\ } {2}}\right)$ which is equivalent to $P\equiv -1\ \left({{\rm mod\ } {2}}\right)$.

See also Wilson's Theorem, Wilson's Theorem Corollary