Suppose . We want to show that and . We have . Reducing , we have . Since , this implies . The other part follows similarily.
Suppose (a,b) = 1. As x runs through a complete residue system (mod b) and y runs through a complete residue system (mod a), then ax + by runs through a complete residue system (mod ab).
I'm curious as to whether this is true. I've tried many examples but have failed to prove the above statement. Any help would be great. Thanks!