MathDB

Fix positive integers

n

and

k\ge 2

. A list of

n

integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add

1

to all of them or subtract

1

from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least

n-k+2

of the numbers on the blackboard are all simultaneously divisible by

k

Problem Statement