Maximum number of sides of a regular polygon

The sum of the interior angles of a regular polygon with $n$ sides is given by the formula $180(n-2)$ . Since there are $n$ interior angles we want $\frac{180(n-2)}{n} = 180 - \frac{360}{n}$ to be an integer which means that $n = \boxed{360}$ .

If the interior angle of a regular polygon is an integer, than the exterior angle will be an integer, too (since both angles add up to the integer $180$ and addition is closed for integers). The exterior angle of a regular polygon with $n$ sides is $\frac{360}{n}$ , so the greatest value of $n$ where $\frac{360}{n}$ is still an integer would be the greatest factor of $360$ , which is $360$ .