Tedious calculation?

If a positive integer $n$ is coprime to $720$ , then $720-n$ is another positive integer that is coprime to $720$ . This pairs up the integers that are coprime to $720$ . (Note that $360$ is not coprime to $720$ .) The arithmetic mean of any such pair is $\frac {n + (720-n)}{2} = 360$ , hence the arithmetic mean of all the numbers coprime to $720$ is $360$ .

Note: There is no need to sum up all of the positive integers that are coprime to $720$ .