MathDB

Let

m,n

and

N

be positive integers and

\mathbb{Z}_{N}=\{0,1,\dots,N-1\}

a set of residues modulo

N

. Consider a table

m\times n

such that each one of the

mn

cells has an element of

\mathbb{Z}_{N}

. A move is choose an element

g\in \mathbb{Z}_{N}

, a cell in the table and add

+g

to the elements in the same row/column of the chosen cell(the sum is modulo

N

). Prove that if

N

is coprime with

m-1,n-1,m+n-1

then any initial arrangement of your elements in the table cells can become any other arrangement using an finite quantity of moves.

Problem Statement