Here’s a diagram that looks like 12.
Can you dissect the red part of the diagram into n congruent pieces for every n that divides 12?
All the angles are multiples of 45° and if two lengths seem equal, they are equal.
Update:
I originally thought that I had an answer for n = 4, but I was mistaken. So I’m turning n = 4 into a bonus question. I’ll still wait for someone to prove or disprove n = 4, but it’s not essential for your answer to be accepted.