Prove that $gcd(n, k) = 1$ implies $n | \binom{n}{k}$

Question

Assume $n$ and $k$ are relatively prime integers. How to prove that $n$ divides $\binom{n}{k}$?

Answer 1

Note that ${n\choose k}{k\choose 1}={n\choose1}{n-1 \choose k-1}$

Hence ${n\choose k}k=n{n-1 \choose k-1}$

Hence $n|{n\choose k}k$. The result follows combining with $n,k$ coprime.