Wilson's Theorem/Corollary 2

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $p$ be a prime factor of $n!$ with multiplicity $\mu$.

Let $n$ be expressed in a base $p$ representation as:

Then:
 * $\dfrac {n!} {p^\mu} \equiv \paren {-1}^\mu a_0! a_1! \dotsb a_m! \pmod p$