Two numbers are called relatively prime if their greatest common divisor is $1$. Grogg's favorite number is the product of the integers from $1$ to $10$. What is the smallest integer greater than $500$ that is relatively prime to Grogg's favorite number?

Product of the integers from $1$ to $10$ is $3628800$.

So, Grogg's favorite number is $3628800$.

The smallest integer greater than $500$ that is relatively prime to Grogg's favorite number should not have a common divisor with $3628800$.

This means, that number should not be divisible by any of the integers from $2$ to $10$.

Clearly, $503$ is the smallest integer greater than $500$ that is relatively prime to Grogg's favorite number.