Generalization of Wilson's Theorem

Theorem

Let $n \in \Z, n > 0$ be a positive integer.

Let $p$ be a prime number.

Let $\displaystyle n = \sum_{j=0}^k a_k p^k$ be the base $p$ presentation of $n$.

Let $p^\mu$ be the largest power of $p$ which divides $n!$, that is:

• $p^\mu \backslash n!$
• $p^{\mu+1} \nmid n!$

Then:

$\displaystyle \frac {n!}{p^\mu} \equiv \left({-1}\right)^\mu a_0! a_1! \ldots a_k! \left({\bmod\, p}\right)$