Show that if a and b are integers with (a,b)=1, then (a+b,a-b)=1 or 2
Should I use the fact that (a+cb,b)=(a,b)? I've also tried to start with (a,b)=1 => 1=xa+yb, but then I was stuck, don't know how to begin. Thanks for helping me out.
Show that if a and b are integers with (a,b)=1, then (a+b,a-b)=1 or 2
Should I use the fact that (a+cb,b)=(a,b)? I've also tried to start with (a,b)=1 => 1=xa+yb, but then I was stuck, don't know how to begin. Thanks for helping me out.
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