Here is my problem:
Let a,b be nonzero integers. Prove that (a,b) = 1 if and only if (a+b, ab) = 1.
Any suggestion on where to start?
You could try playing around with Bezout's identity.
(From right to left) Suppose that . There exist integers such that . Distributing the terms, , and . This shows that must divide 1. Therefore, .
You can try a similar argument for the other direction. (You may need to insert some terms into the equation by adding and subtracting the same thing.)