jmc

By the Euclidean Algorithm: $$\begin{array}{rl}\text{gcd}(40304,30203)& =\text{gcd}(40304-30203,30203)\\ & =\text{gcd}(10101,30203)\\ & =\text{gcd}(30203-2\cdot 10101,10101)\\ & =\text{gcd}(10001,10101)\\ & =\text{gcd}(10101-10001,10001)\\ & =\text{gcd}(100,10001)\\ & =\text{gcd}(10001-100\cdot 100,100)\\ & =\text{gcd}(1,100)\end{array}$$Therefore, the greatest common divisor of $40304$ and $30203$ is $\boxed{1}$