NO-ABEL 2007 Problem 4

Let $a, b$ and $c$ be integers such that $a + b + c = 0$ . (a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$ . (b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$ . .