jmc

We can use the Euclidean Algorithm. The closest multiple of $2a+9$ that we can spot to $6a^2+49a+108$ is $6a^2+49a+99=(2a+9)(3a+11),$ so we have $$\begin{array}{rl}\text{gcd}(6a^2+49a+108,2a+9)& =\text{gcd}(6a^2+49a+108-(2a+9)(3a+11),2a+9)\\ & =\text{gcd}(6a^2+49a+108-(6a^2+49a+99),2a+9)\\ & =\text{gcd}(9,2a+9).\end{array}$$Since $7767$ is a multiple of 9, both $2a$ and $9$ are multiples of $9$, $2a+9$ is also a multiple of $9$ so the greatest common divisor is $\boxed{9}$.