IMO 1967 Problem 3

Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$ . Let $c_s=s(s+1)$ . Prove that

$$(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)$$

is divisible by the product $c_1c_2\ldots c_n$ .