Wilson's Theorem/Corollary 2

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