On the Greatest Common Divisor of Binomial Coefficients $\left(\genfrac{}{}{0ex}{}{n}{q}\right),\left(\genfrac{}{}{0ex}{}{n}{2q}\right),\left(\genfrac{}{}{0ex}{}{n}{3q}\right),\dots$

Carl McTague

Every binomial coefficient aficionado knows that the greatest common divisor of the binomial coefficients $\left(\genfrac{}{}{0ex}{}{n}{1}\right),\left(\genfrac{}{}{0ex}{}{n}{2}\right),\dots ,\left(\genfrac{}{}{0ex}{}{n}{n-1}\right)$ equals $p$ if $n=p^i$ for some $…