Any rotation of a regular polygon with n sides about its center 
through an angle which is a positive integer multiple of the 
central angle of the regular polygon carries the regular polygon 
onto itself.  The central angle of a regular polygon with n sides 
is 360°/n.  

For a regular pentadecagon, the central angle is 360°/15 = 24°
For a regular icosagon, the central angle is 360°/20 = 18°

The smallest positive angle of rotation about their centers that 
carries both a regular pentadecagon onto itself and a regular 
icosagon onto itself is therefore the least common multiple of 
24° and 18°, which is 72°.

Edwin