Wilson's Theorem/Corollary 2

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

Let $$p$$ be a prime number.

Let $$n = \sum_{j=0}^k a_k p^k$$ 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:
 * $$\frac {n!}{p^\mu} \equiv \left({-1}\right)^\mu a_0! a_1! \ldots a_k! \left({\bmod\, p}\right)$$