Let a, b, n be integers such that 0 < a ≤ b < n. Prove that there
exists a prime number p that divides both
\begin{equation}\binom{n}{b}, \binom{n}{a}\end{equation}
I have been proving with the next identity of the binomial coefficients:
\begin{equation}\binom{n}{b}\cdot\binom{b}{a}=\binom{n}{a}\cdot\binom{n-a}{b-a}\end{equation}
I think that this idea can help to reach the solution. I would be grateful if someone could get any easy solution.
Best Answer
If $\binom{n}{a}$ and $\binom{n}{b}$ were relatively prime, the identity gives that $\binom{n}{a}\mid\binom{b}{a}$, in particular $n<b$. But we assumed $b<n$. So they must have a gcd.