MathNet 2013 Question 18

Determine all positive integers $n$ such that $f_n(x, y, z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x, y, z) = (x-y)^{5n} + (y-z)^{5n} + (z-x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.