ELMO 2013 Problem 5

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$ , the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$ ?

Proposed by Andre Arslan