Is it possible to divide a circle into $7$ equal "pizza slices" (using geometrical methods)? Or is it possible to divide a circle into n equal "pizza slices" (I don't know how to call these parts, but I think you'll know what I mean), where n hasn't got a common divider with $360$? Or are the $360$ degrees just "arbitrarily" chosen, in such way that it's also possible to make a system with $7$ "degrees" in a circle?
The main question is actually if it's possible with a ruler and a pair of compasses to divide a circle into any number of slices, and if there's a condition for a number (e.g. a common divider with $360$ as I suggested) so it's possible to slice the circle in such a number of pieces.

The degree units have nothing to do with constructability using ruler and compasses.
The regular $n$- gon can be constructed if and only if $n$ is the product of a power of two and zero or more distinct odd primes of the form $p=2^k+1$. There are not so many such primes (called Fermat primes), namely $3, 5, 17, 257, 65537$ - and as far as is known today no more. Thus it is possible to construct the $6$-gon and the $8$-gon, but not the $7$-gon and not the $9$-gon. So I suggest you invite another (single!) friend to your pizza. 

Exercise: If $2^k+1$ is prime then $k$ is a power of two.
