ILL 1982 Problem 1

(a) Prove that $\frac{1}{n+1} \cdot \binom{2n}{n}$ is an integer for $n \geq 0.$ (b) Given a positive integer $k$ , determine the smallest integer $C_k$ with the property that $\frac{C_k}{n+k+1} \cdot \binom{2n}{n}$ is an integer for all $n \geq k.$