As A Whole...

The sum of the interior angles of an $n$ -gon, in degrees, is $180(n-2)$ . The value of the interior angles of a regular $n$ -gon is thus $\theta=\frac{180(n-2)}{n}=180(1-\frac{2}{n})=180-\frac{360}{n}$ In order for $\theta$ to have an integer value, $n\vert360$ . The prime factorization of $360$ is $360=2^3\times3^2\times5$ There are thus $(3+1)(2+1)(1+1)=24$ distinct factors of $360$ . Discounting the degenerate values of $n=1,2$ , we have that there are $\large{24-2=\boxed{22}}$ possible regular $n$ -gons with integer interior angles, in degrees.

Interior angle is integer $\Rightarrow$ exterior angle is integer

Exterior angles of n-side regular polygon are all equal to $\frac{360}{n}$ ...... $(1)$

This means $n$ is factor of $360$

$360=2^3×3^2×5$ ,

Therefore $(3+1)(2+1)(1+1)=24$ factors but out of these, $180$ and $360$ have to be excluded as then $(1)=2,1$ and polygon must have atleast 3 sides.

Therefore answer $=24-2=\boxed{\boxed{22}}$

Factors are $1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360$