MathNet 2014 Question 69

For any integer $n$, let $M(n) = \{n, n+1, n+2, n+3, n+4\}$. Let $S(n)$ denote the sum of the squares of all elements of $M(n)$ and let $P(n)$ denote the product of these squares. For which integers $n$ is $S(n)$ a divisor of $P(n)$?