Binomial Coefficient Divisibility

Question

I'd like to ask whether there exists some simple criterion to determine whether the binomial coefficient $\binom{n}{m}$ is divisible by a prime $p$. For the case $p=2$, the criterion is $\binom{n}{m} \equiv 0 \ (mod \ 2) \iff n \ \& \ m \neq m$, where $\&$ is the bitwise-and operation.