ok - so after a long proof that I know is correct, I have come to conclude that:
given: gcd(2a+b, a+2b) = 1 or 2 or 3
I have to prove that this does not equal 2.
I also know (and have proven) that gcd(a,b)=1.
I know I need to show that for d to not equal 2, I would have to pretend and show that when d equals 2, d could not equal 1 - which is false since d has to equal 1.
So to start: Assume gcd(2a+b, a+2b) = 2, then it divides both a and b