For every $n \in \mathbb{N}$ $\pfac{n}$ stands for the product of numbers between $1$ and $n$ which are not divisible by a given prime $p$ . And we set $\pfac{0} =1$ .

The corollary below generalizes a result first found by Anton, Stickelberger, and Hensel:

Let $N_0$ be the least non-negative residue of $n \pmod{p^s}$ where $p$ is a prime number and $n \in \mathbb{N}$ . Then

Proof. We write each $r$ in the product below as $ip^s +j$ to get \begin{eqnarray*} \pfac{n} &=& \prod\limits_{\substack{1 \le r \le n\\ p^s \not\div r}} r \\ &=&\left( \prod\limits_{\substack{0 \le i \le \left\lfloor n/p^s\right\rfloor -1 \\ 1 \le j < p^s \\ p^s \not\div j}} ip^s +j\right)\left( \prod\limits_{\substack{i=\left\lfloor n/p^s\right\rfloor \\ 1\le j \le N_0 \\ p^s \not\div j}} ip^s +j\right) \\ &\equiv& \prod\limits_{i=0}^{\left\lfloor n/p^s \right\rfloor -1} \prod\limits_{\substack{1 \le j < p^s \\ p^s \not\div j }} j\cdot \prod\limits_{\substack{j=1 \\ p^s \not\div j}}^{N_0} j) \\ &\equiv& \pfac{p^s}^{\left\lfloor n/p^s\right\rfloor}\cdot \pfac{N_0} \pmod{p^s}. \end{eqnarray*}From Wilson's theorem for prime powers it follows that