MathDB

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)

.
Proposed by Boldizsár Varga, Budapest

Problem Statement