  We first prove that if $a$ has a prime factor not dividing $b$ then there are
infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not
divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover,
we provide some new divisibility properties of binomial coefficients: for
example, we prove that $\binom {12n} {3n}$ and $\binom {12n} {4n}$ are
divisible by $6n-1$, and that $\binom {330n} {88n}$ is divisible by $66n-1$,
for all positive integers $n$. As we show, the latter results are in fact
consequences of divisibility and positivity results for quotients of
$q$-binomial coefficients by $q$-integers, generalizing the positivity of
$q$-Catalan numbers. We also put forward several related conjectures.
