jmc

We use the Euclidean algorithm. $$\begin{array}{rl}\text{gcd}(2863,1344)& =\text{gcd}(2863-1344\cdot 2,1344)\\ & =\text{gcd}(175,1344)\\ & =\text{gcd}(175,1344-175\cdot 7)\\ & =\text{gcd}(175,119)\\ & =\text{gcd}(175-119,119)\\ & =\text{gcd}(56,119)\\ & =\text{gcd}(56,119-56\cdot 2)\\ & =\text{gcd}(56,7).\end{array}$$Since $56$ is a multiple of $7$, the greatest common divisor is $\boxed{7}$.