I'm curious about the following elementary question.
Suppose that $a_1,a_2,b_1,b_2$ are integers such that $\mathrm{gcd}(a_1,b_1)=\mathrm{gcd}(a_2,b_2)=1$ and either $a_1\neq a_2$ or $b_1\neq b_2$ (or both).
Question: Assume in addition that $\mathrm{gcd}(a_1+a_2,b_1+b_2)=1$ holds.
What does this imply for the integers $a_1,a_2,b_1,b_2$?
In particular, do they have to satisfy the following relation $a_1b_2-a_2b_1=\pm 1$ (or a similar relation)? Edit: There are simple counterexamples to this (as in the comment by Mr.Brooks), so this relation does not necessarily hold.
Thanks for any help, references are also much appreciated.
Note: I've edited the original question, which was: When is $\mathrm{gcd}(a_1+a_2,b_1+b_2)=1$?.