Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. Sun

For all positive integers n, we prove the following divisibility properties: $$(2n+3)\left(\genfrac{}{}{0px}{}{2n}{n}\right)\mathrm{\mid }3\left(\genfrac{}{}{0px}{}{6n}{3n}\right)\left(\genfrac{}{}{0px}{}{3n}{n}\right),and(10n+3)\left(\genfrac{}{}{0px}{}{3n}{n}\right)\mathrm{\mid }21\left(\genfrac{}{}{0px}{}{15n}{5n}\right)\left(\genfrac{}{}{0px}{}{5n}{n}\right)\mathrm{.}$$ 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.