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) ( 2n n ) ∣∣∣∣3(6n 3n )( 3n n ) and (10n + 3) ( 3n n ) ∣∣∣∣21(15n 5n )( 5n 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 am )( an+bn 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.