MathNet 2015-part-05 — 191 problems (EN–VI) Question 115

Let $k$ be a natural number. For $n \in \mathbb{N}$ denote by $f_{k}(n)$ the smallest natural number greater than $k n$ such that $n f_{k}(n)$ is a perfect square of a natural number. If $f_{k}(m)=f_{k}(n)$, prove that $m=n$.