IMC 2011 Problem 3

Let $p$ be a prime number. Call a positive integer $n$ interesting if $$x^n-1=(x^p-x+1)f(x)+pg(x)$$ for some polynomials $f$ and $g$ with integer coefficients. a) Prove that the number $p^p-1$ is interesting. b) For which $p$ is $p^p-1$ the minimal interesting number?