Putnam 1977 A5

Problem A-5 Prove that $$\binom{p a}{p b} \equiv\binom{a}{b}(\bmod p)$$ for all integers $p, a$, and $b$ with $p$ a prime, $p>0$, and $a>b>0$. Notation: $\binom{m}{n}$ denotes the binomial coefficient $\frac{m!}{n!(m-n)!}$.