International Mathematical Olympiad 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 the product $$(c_{m+1} - c_k)(c_{m+2} - c_k) \cdots (c_{m+n} - c_k)$$ is divisible by the product $c_1c_2\cdots c_n$.