Lucas' Theorem

Theorem
Let $p$ be a prime number.

Let $n, k \in \Z$.

Then:
 * $\displaystyle \binom n k \equiv \binom {\left \lfloor {n / p} \right \rfloor} {\left \lfloor {k / p} \right \rfloor} \binom {n \bmod p} {k \bmod p} \pmod p$

where:
 * $\displaystyle \binom n k$ is a binomial coefficient
 * $n \bmod p$ denotes the modulo operation
 * $\left \lfloor \cdot \right \rfloor$ is the floor function.

Hence, if the representations of $n$ and $k$ to the base $p$ are given by:
 * $n = a_r p^r + \cdots + a_1 p + a_0$
 * $k = b_r p^r + \cdots + b_1 p + b_0$

then:
 * $\displaystyle \binom n k \equiv \prod_{j \mathop = 0}^r \binom {a_j} {b_j} \pmod p$

Proof of first part
First we show that:
 * $\displaystyle \binom n k \equiv \binom {\left \lfloor {n / p} \right \rfloor} {\left \lfloor {k / p} \right \rfloor} \binom {n \bmod p} {k \bmod p} \pmod p$

Consider $\displaystyle \binom n k$ as the fraction:
 * $\dfrac {n \left({n-1}\right) \left({n-2}\right) \cdots \left({n-k+1}\right)} {k \left({k-1}\right) \left({k-2}\right) \cdots 1}$

This can be expressed as:
 * $\displaystyle (1): \quad \binom n k = \left({\frac n k}\right) \left({\frac {n-1}{k-1}}\right) \left({\frac {n-2}{k-2}}\right) \cdots \left({\frac {n-k+1}{1}}\right)$

Let $k = s p + t$ from the Division Theorem.

Thus:
 * $t = k \bmod p$

The denominators of the first $t$ factors in $(1)$ do not have $p$ as a divisor.

Now let $n = u p + v$, again from the Division Theorem.

Thus:
 * $v = n \bmod p$

Now, when dealing with non-multiples of $p$, we can work modulo $p$ in both the numerator and denominator, from Common Factor Cancelling in Congruence.

So we consider the first $t$ factors of $(1)$ modulo $p$:

These are:
 * $\left({\dfrac {u p + v}{s p + t}}\right) \left({\dfrac {u p + v-1}{s p + t-1}}\right) \cdots \left({\dfrac {u p + v-t+1}{s p + 1}}\right) \equiv \left({\dfrac v t}\right) \left({\dfrac {v-1}{t-1}}\right) \cdots \left({\dfrac {v-t+1} 1}\right) \pmod p$

So, these first $t$ terms of $(1)$ taken together are congruent modulo $p$ to the corresponding terms of:
 * $\displaystyle \binom {n \bmod p} {k \bmod p}$

These differ by multiples of $p$.

So we are left with $k - k \bmod p$ factors.

These fall into $\left \lfloor {k / p} \right \rfloor$ groups, each of which has $p$ consecutive values.

Each of these groups contains exactly one multiple of $p$.

The other $p-1$ factors in a given group are congruent (modulo $p$) to $\left({p-1}\right)!$ so they cancel out in numerator and denominator.

We now need to investigate the $\left \lfloor {k / p} \right \rfloor$ multiples of $p$ in the numerator and denominator.

We divide each of them by $p$ and we are left with the binomial coefficient:
 * $\displaystyle \binom{\left \lfloor {\left({n - k \bmod p}\right) / p} \right \rfloor} {\left \lfloor {k / p} \right \rfloor}$

Now, if $k \bmod p \le n \bmod p$, this equals:
 * $\displaystyle \binom {\left \lfloor {n / p} \right \rfloor} {\left \lfloor {k / p} \right \rfloor}$

Otherwise, if $k \bmod p > n \bmod p$, the other factor:
 * $\displaystyle \binom {n \bmod p} {k \bmod p}$

is zero.

So the formula holds in general.

Proof of second part
Now we consider the representations of $n$ and $k$ to the base $p$:
 * $n = a_r p^r + \cdots + a_1 p + a_0$
 * $k = b_r p^r + \cdots + b_1 p + b_0$

Let $n_1 = \left \lfloor {n / p} \right \rfloor, k_1 = \left \lfloor {k / p} \right \rfloor$.

We have that:
 * $n \bmod p = a_0, k \bmod p = b_0$
 * $n_1 = a_r p^{r-1} + a_{r-1} p^{r-2} \cdots + a_1, k_1 = b_{r-1} p^{r-2} \cdots + b_1$

From the proof of the first part, this gives us:
 * $\displaystyle \binom n k \equiv \binom {n_1} {k_1} \binom {a_0} {b_0} \pmod p$

Now we do the same again to the representation to the base $p$ of $n_1$ and $n_2$.

Thus:
 * $\displaystyle \binom n k \equiv \binom {\left \lfloor {n_1 / p} \right \rfloor} {\left \lfloor {k_1 / p} \right \rfloor} \binom {a_1} {b_1} \binom {a_0} {b_0} \pmod p$

and so on until $\left \lfloor {n_r / p} \right \rfloor$ and $\left \lfloor {k_r / p} \right \rfloor$.

Hence the result.

It appears in his 1878 paper Théorie des Fonctions Numériques Simplement Périodiques in the American Journal of Mathematics Volume 1, issues 2 - 4.