combinatorics - How to prove that $n$ divides $binom{n}{k}$ if $n$ is prime?

I want to prove that n divides $\binom{n}{k}$ and so I expanded the term to
$\frac{n(n-1)..(n-k+1)}{k!}$. Clearly $n$ divides the numerator and also $n$ is relatively prime to all of the terms in the denominator and so $n$ is not divisible by $k!$. I'm struggling with how to approach that $\frac{(n-1)..(n-k+1)}{k!}$ is integer.

This problem comes as an example in the book I'm reading and supposedly it's obvious but I don't see it.

Edit: Sorry adding that we must have $1 \leq k \leq n-1$.