1976 AHSME Problem 23

For integers k and n such that 1\le k<n, let C^n_k=\frac{n!}{k!(n-k)!}. Then \left(\frac{n-2k-1}{k+1}\right)C^n_k is an integer

\begin{array}{l} \textbf{(A) }\text{for all }k\text{ and }n\qquad \\ \textbf{(B) }\text{for all even values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad \\ \textbf{(C) }\text{for all odd values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad \\ \textbf{(D) }\text{if }k=1\text{ or }n-1,\text{ but not for all odd values }k\text{ and }n\qquad \\ \textbf{(E) }\text{if }n\text{ is divisible by }k,\text{ but not for all even values }k\text{ and }n \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.