I'm stuck on this and am not sure about the right way to proceed; what I did ended up being entirely wrong so I won't post it -at the time, our professor touched on it and it was entirely different from what I did. I really want to know how to correctly do this problem and I am not able to receive relevant help on it now outside of this site so I would be very appreciative of insight.

Prove that

$$(n+1)\mid{{2n}\choose{n}}$$ for every $n \in\mathbb N$.

There was a suggestion to write the quotient as a difference of binomial coefficients.

Thanks