Question 81
Let $M, N$ be two positive integers. We put a light on each unit square of the table $N \times N$ (each light is either on or off). Initially, all the lights are off. In each step, we choose either a column or a row and switch $M$ consecutive lights on the chosen column/row. Find the necessary and sufficient condition of $M, N$ such that we can switch on all the lights after finite steps.