  For all positive integers n, we prove the following divisibility properties:
$$(2n+3){2n\choose n} | 3{6n\choose 3n}{3n\choose n}, and (10n+3){3n\choose n}
| 21{15n\choose 5n} {5n\choose n}.$$ This confirms two recent conjectures of
Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show
that, for all positive integers m and n, the product $am{am+bm-1\choose
am}{an+bn\choose an}$ is divisible by m+n. In fact, the latter result can be
generalized to the q-binomial coefficients and q-integers case, which
generalizes the positivity of q-Catalan numbers. We also propose several
related conjectures.
