TITLE: When does $r$ divide $\binom{n}{r}$?
QUESTION [7 upvotes]: Suppose $n>r$ and both $n$ and $r$ are positive integers. 
When can we have that $\dbinom{n}{r}$ is divisible by $r$ ?
I'm looking for all pairs $(n, r)$ such that the above condition is satisfied.

REPLY [5 votes]: I think this question is quite difficult. The answer for the case that $r$ is prime can be deduced from Lucas' theorem, which says in particular that ${n\choose r}$ is divisibly by a prime $p$ if and only if one of the digits of $r$ when written in base $p$ is greater than the corresponding base $p$ digit of $n$.
Since $r=10_r$ written in base $r$, we obtain that

If $r$ is prime then ${n\choose r}$ is divisible by $r$ if and only if the remainder of $n$ modulo $r^2$ is smaller than $r$.