jmc

If $a=b=40$, then $a+b=80$, and $\gcd (a,b)=\gcd (40,40)=40$. If one of $a$ and $b$ is greater than 40, then the other is less than 40, in which case $\gcd (a,b)$ must also be less than 40. Therefore, the largest possible value of $\gcd (a,b)$ is $\boxed{40}$.