To find the greatest common divisor of $50$ and $5005$, I can use the Euclidean algorithm, which repeatedly applies the division algorithm to find remainders and quotients. So, I start by dividing $5005$ by $50$, and I get a quotient of $100$ and a remainder of $5$. That means that the greatest common divisor of $50$ and $5005$ is the same as the greatest common divisor of $50$ and $5$. Now, I divide $50$ by $5$, and I get a quotient of $10$ and a remainder of $0$. That means that the greatest common divisor of $50$ and $5$ is $5$, and there are no more remainders to check. So, the greatest common divisor of $50$ and $5005$ is $5$. To find the least common multiple of $50$ and $5005$, I can use the formula that the least common multiple of two numbers is equal to their product divided by their greatest common divisor. So, the least common multiple of $50$ and $5005$ is $50 \times 5005 \div 5$. That simplifies to $10 \times 5005$, which is $50050$. Now, to find the sum of the greatest common divisor and the least common multiple, I just add them together. So, the sum is $5 + 50050$, which is $50055$. 
What is the sum of the greatest common divisor of $50$ and $5005$ and the least common multiple of $50$ and $5005$?