16. A polygon is said to be friendly if it is regular and it also has angles that when measured in degrees are either integers or half-integers (i.e. have a decimal part of exactly 0.5 ). How many different friendly polygons are there?
SolUTION
028

The interior angle will be an integer or half-integer precisely when the exterior angle is an integer or a half-integer respectively. A regular polygon of $n$ sides has exterior angle $\frac{360}{n}$. This is either an integer or a half-integer if, and only if, $\frac{720}{n}$ is an integer. Therefore, $n$ must be a factor of 720 . The factors of 720 are $720,360,240,180,144,120,90,80,72,60,48,45,40$, $36,30,24,20,18,16,15,12,10,9,8,6,5,4$ and 3 (ignoring 2 and 1 neither of which is a valid number of sides for a polygon). There are 28 numbers in this list, so there are 28 different friendly polygons.