NL-NWO 1965 Problem 2

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$ . Prove that for no $n$ : $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$ . Prove that $S_1 \cdot S_2$ is divisible by $150$ .