jmc

We use the Euclidean algorithm to find the greatest common divisor. $$\begin{array}{rl}\text{gcd}(10293,29384)& =\text{gcd}(29384-2\cdot 10293,10293)\\ & =\text{gcd}(8798,10293)\\ & =\text{gcd}(8798,10293-8798)\\ & =\text{gcd}(8798,1495)\\ & =\text{gcd}(8798-1495\cdot 5,1495)\\ & =\text{gcd}(1323,1495)\\ & =\text{gcd}(1323,1495-1323)\\ & =\text{gcd}(1323,172)\\ & =\text{gcd}(1323-172\cdot 7,172)\\ & =\text{gcd}(119,172)\\ & =\text{gcd}(119,172-119)\\ & =\text{gcd}(119,53)\\ & =\text{gcd}(119-53\cdot 2,53)\\ & =\text{gcd}(13,53).\end{array}$$At this point we can see that since $53$ is not divisible by the prime number $13$, the greatest common divisor is just $\boxed{1}$.