jmc

The prime factorization of 50 is $2\cdot 5^2$ and the prime factorization of 5005 is $5\cdot 7\cdot 11\cdot 13$. The greatest common divisor is 5 and the least common multiple is $2\cdot 5^2\cdot 7\cdot 11\cdot 13=2\cdot 5\cdot 5005=50050$. The sum of the GCD and LCM is $\boxed{50055}$.