Question 39
Let $p$ be a prime number and let $m$ and $n$ be positive integers written in base $p$ as $n = a_0 + a_1p + \dots + a_kp^k$ and $m = b_0 + b_1p + \dots + b_kp^k$, respectively. Show that
$$
\binom{n}{m} \equiv \prod_{i=0}^{k} \binom{a_i}{b_i} \pmod{p}
$$
$$
\binom{n}{m} \equiv \prod_{i=0}^{k} \binom{a_i}{b_i} \pmod{p}
$$