KöMaL 2024 A886

Let $k$ and $n$ be two given distinct positive integers greater than 1. There are finitely many (not necessarily distinct) integers written on the blackboard. Kázmér is allowed to erase $k$ consecutive elements of an arithmetic sequence with a difference not divisible by $k$. Similarly, Nándor is allowed to erase $n$ consecutive elements of an arithmetic sequence with a difference that is not divisible by $n$. The initial numbers on the blackboard have the property that both Kázmér and Nándor can erase all of them (independently from each other) in a finite number of steps. Prove that the difference of biggest and the smallest number on the blackboard is at least $\varphi(n)+\varphi(k)$, where $\varphi$ denotes Euler's totient function, i.e., $\varphi(n)$ is the number of positive integers not exceeding $n$ that are coprime to $n$.