Prove that if for , then .
Let be any common divisor of and . Then we have and therefore Similarly, let be any common divisor of and . Then we have and therefore How does it follow that and have the same common divisors?
When you wrote "therefore you should also notice that (by hypothesis). The same thing for . You should also note that (by hypothesis).
Now what you have is that if is a common divisor of and , then it's also of and . Similarly, if is a common divisor of and , then it's also of and .
Therefore, the pair has the same common divisors as the pair . In particular, the greatest common divisor.