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.
