Explaining how the accepted solution works of a previous problem

Question

• https://math.stackexchange.com/a/769735/1021792 in the above stack post which was accepted as a answer , what will number $a_j - a_i$ indicates so as to solve the problem?

What i get is that $a_1$ divides $a_j-a_i$ , but this doesnt tell me anything more . How to show there are two numbers that one evenly divides the other wih this approach .

• Note : i know the other proof of n boxes i would like to see how the accepted answer show us the proof of the claim . ( Also it should be $a_1 \leq 100$ not $99$)