Well, the "2" case is easy. Consider two integers a, b such that (a,b) = 1 and both a and b are odd. Thus a+b and a-b are both even, so (a+b,a-b) = 2.

Can't figure anything out for the other case, where a is even and b is odd. (Obviously this is the same as the case for odd a and even b. And, of course, we can't have both a and b even.)

-Dan