Freaks City

Lucas Theorem Mathematical Statement Given prime $p$, the following congruence holds: $$\binom{n}{m} \equiv \binom{\lfloor n/p \rfloor}{\lfloor m/p \rfloor} \cdot \binom{n \bmod p}{m \bmod p} \pmod{p}$$ Proof Foundation Lemma 1 For prime $p$: $$\binom{p}{k} \equiv 0 \pmod{p} \text{ when } 0 < k < p$$ The numerator contains factor $p$, mak ...