BALKAN 2010 Problem 4

For every integer $n \geq 2$, denote by $f(n)$ the sum of positive integers not exceeding $n$ that are not coprime to $n$. Prove that $f(n+p) \neq f(n)$ for any such $n$ and any prime number $p$. (Turkey)

Time allowed: 270 minutes. Each problem is worth 10 points.