CIIM 2021 Problem 3

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.