More Divisibility Properties for Binomial Coefficients

Abstract:

      Let $a$, $b$ and $n$ be positive integers with $a>b$, we prove the following divisibility property: For all positive integers $n$, we have $$(2bn+1)(2bn+3)(2bn+5){2bn\choose bn}\Big|15(a-b)(3a-b)(5a-b)(5a-3b){2an \choose an}{an\choose bn},$$ which extends the result of Yang. And for all positive integers $n$, we show the following divisibility properties: $$(6n+1){4n\choose n}\Big|{12n\choose 6n}{2n\choose n},\ (12n+1){5n\choose n}\Big|{15n\choose 3n}{3n-1\choose n-1},$$ $$(18n+1){12n\choose 9n}{8n\choose 2n}\Big| {24n\choose 18n}{4n\choose 2n}{6n\choose 3n}.$$ Other more similar divisibility properties are given also.