AUSTRIAN-POLISH 2006 Problem 4

A positive integer $d$ is called nice iff for all positive integers $x,y$ hold: $d$ divides $(x+y)^{5}-x^{5}-y^{5}$ iff $d$ divides $(x+y)^{7}-x^{7}-y^{7}$ . a) Is 29 nice? b) Is 2006 nice? c) Prove that infinitely many nice numbers exist.