Does $ p|(2^{2kq}-2^{kq}+1)$ where $p=1+k\cdot q$ ? I'm stuck...

Question

Does $ p|(2^{2kq}-2^{kq}+1)$ ,$p=1+k\cdot q$,where $p,q$ are prime ?

From Fermat's little theorem;
$(2^{2kq}-2^{kq}+1)$ mod $q\equiv (2^{2k}-2^{k}+1)$

This is where I'm stuck, please help.

Thank you...

Answer 1

You have that $$2^{p-1}\equiv 1\equiv 2^{kq}\pmod{p}$$