For non-negative integers $m$ and $n$ and a prime $p$, the following congruence relation holds:
$$\binom{m}{n} \equiv \prod_{i=0}^{k} \binom {m_i}{n_i} \pmod {p},$$
where
$$m=m_{k}p^{k}+m_{k-1}p^{k-1}+\cdots +m_{1}p+m_{0},$$
and
$$n=n_{k}p^{k}+n_{k-1}p^{k-1}+\cdots +n_{1}p+n_{0},$$
are the base $p$ expansions of $m$ and $n$ respectively. This uses the convention that $ \tbinom {m}{n}=0$ if $m < n$.
推论
A binomial coefficient $\tbinom {m}{n}$ is divisible by a prime $p$ if and only if at least one of the base $p$ digits of $n$ is greater than the corresponding digit of $m$.
Lucas 定理的递归形式
对于非负整数 $m$，$n$ 和质数 $p$ 设 $m = ap + q$，$n = bp + r$（$0\le q, r < p$ ）则有
$$ \binom{m}{n} \equiv \binom{a}{b} \binom{q}{r} \pmod{p} $$
递归边界是 $ \tbinom{m}{0} = 1$ 和 $\binom{m}{n} = 0$ 若 $m < n$ 。