A characterization of a prime $p$ from the binomial coefficient $\left(\genfrac{}{}{0px}{}{n}{p}\right)$

We complete a proof of a theorem that was inspired by an Indian Olympiad problem, which gives an interesting characterization of a prime number $p$ with respect to the binomial coefficients $\left(\genfrac{}{}{0px}{}{n}{p}\right)$. We also derive a related result which generalizes the theorem in one direction.