Let $k$ be a natural number. For each natural number $n$ we define $f_k (n)$ to be the least number, greater than $kn$, for which $nf_k (n)$ is a perfect square. Prove that $f_k (n)$ is injective.
Problem
Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade
Tags: number theory, injective function, function