jmc

We use the Euclidean algorithm. $$\begin{array}{rl}\text{gcd}(5616,11609)& =\text{gcd}(5616,11609-2\cdot 5616)\\ & =\text{gcd}(5616,377)\\ & =\text{gcd}(5616-14\cdot 377,377)\\ & =\text{gcd}(338,377)\\ & =\text{gcd}(338,377-338)\\ & =\text{gcd}(338,39)\\ & =\text{gcd}(338-8\cdot 39,39)\\ & =\text{gcd}(26,39).\end{array}$$We can easily find that the greatest common divisor of $26$ and $39$ is $\boxed{13}$.