Abstract:

In this paper, we prove that if $n\geq4$ and $a\ge 0$ are integers satisfying $a<\frac{n}{3}$, then

$$\gcd\left(\left\{\binom{n}{k}:a<k<n-a\right\}\right)=\prod_{n=p^{m}+b(n,p),\ 0\le b(n,p)\leq a,} p,$$ where $\binom{n}{k}=\frac{n!}{k!(n-k)!}$, and the product in the right hand side runs through all primes $p$ such that $ n=p^{m}+b(n,p), m\in\mathbb{N}$ and $0\le b(n,p)\leq a$.

As an application of our result, we give an answer to a problem in Hong [16].

Key words: Binomial coefficient , Greatest common divisor