KMO 2008 Problem 6

There is $n\times n$ chessboard. Each square has a number between $0$ and $k$ . There is a button for each row and column, which increases the number of $n$ numbers of the row or column the button represents(if the number of the square is $k$ , then it becomes $0$ ). If certain button is pressed, call it 'operation.'

And we have a chessboard which is filled with 0(for all squares). After some 'operation's, the numbers of squares are different now. Prove that we can make all of the number $0$ within $kn$ 'operation's.