This question is related to this other question (asked by user @GeraltofRivia). In fact it is what I understood the question to be before it was edited.
Let $n\in\mathbb{N}$. For what $1\le r\le n-1$ is $\binom{n}{r}$ divisible by $n$?
When $n$ is prime, this is obviously true for all such $r$. I believe that the case where $n$ is the product of two primes or a power of a single prime shouldn't be too hard to treat, however the general case looks like a combinatorial mess to me. Does anyone have some insight on how the problem can be solved for arbitrary $n$?
