Proving pCr is divisible by p where p is prime and 1 ≤ r ≤p-1 by induction

Question

There is solution of this problem. But i am trying to prove this by using induction and here is my try.
Initial step: The statement is true for r=1, as $\binom{p}{r}=\binom{p}{1}=p$.
Induction step:Assume statement is true for some $1 \le k \le p -1$, that is $$\binom{p}{k} = mp$$ for some $m \in \mathbb Z$. Then for r = k+1 we have $$\binom{p}{k+1} = \frac{p-k}{k+1} \binom{p}{k}$$ $$ \binom{p}{k+1}= mp \frac{p-k}{k+1}$$ by induction hypothesis.
At this point i don't know how can i use the assumption that p is prime to show $\frac{m(p-k)}{k+1}$ is an integer. I need hint. Thank you.