Putnam 1982 B4

Let $n_1,n_2,\ldots,n_s$ be distinct integers such that $$(n_1+k)(n_2+k)\cdots(n_s+k)$$ is an integral multiple of $n_1n_2\cdots n_s$ for every integer $k$ . For each of the following assertions give a proof or a counterexample: $(\text a)$ $|n_i|=1$ for some $i$ $(\text b)$ If further all $n_i$ are positive, then $$\{n_1,n_2,\ldots,n_2\}=\{1,2,\ldots,s\}.$$