Problem

Source: III International Festival of Young Mathematicians Sozopol 2012, Theme for 10-12 grade

Tags: number theory, Eulers function



For a natural number $x$ we define $f(x)$ to be the sum of all natural numbers less than $x$ and coprime with it. Let $m$ and $n$ be some natural numbers where $n$ is odd. Prove that there exist $x$, which is a multiple of $m$ and for which $f(x)$ is a perfect n-th power.