### Proof of two Divisibility Properties of Binomial Coefficients Conjectured by Z.-W. Sun

#### Abstract

For all positive integers $n$, we prove the following divisibility properties:

\[ (2n+3){2n\choose n} \left|3{6n\choose 3n}{3n\choose n},\right. \quad\text{and}\quad

(10n+3){3n\choose n} \left|21{15n\choose 5n}{5n\choose n}.\right. \]

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 further generalized to the $q$-binomial coefficients and $q$-integers case, which generalizes the positivity of $q$-Catalan numbers. We also propose several related conjectures.

\[ (2n+3){2n\choose n} \left|3{6n\choose 3n}{3n\choose n},\right. \quad\text{and}\quad

(10n+3){3n\choose n} \left|21{15n\choose 5n}{5n\choose n}.\right. \]

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 further generalized to the $q$-binomial coefficients and $q$-integers case, which generalizes the positivity of $q$-Catalan numbers. We also propose several related conjectures.

#### Keywords

Congruences; Binomial coefficients; $p$-Adic order; $q$-Catalan numbers; Reciprocal and unimodal polynomials