Is $L$ always context free? Consider formal language $L$ over finite alphabet $\Sigma$ consisting of all words over $\Sigma$ that have non-trivial period (non empty prefix that is also a suffix). Is $L$ always context free?


Maybe pumping lemma will do? I was advised to try with a word $a^Nb^Na^Nb^N$ but if I pump only second block of $a$ then this word still is in $L$, because it has non empty prefix $a^Nb^N$ that is also a suffix.
Try to use closure properties instead.

Closer hint:


   CFL is closed against (pre)image of language homomorphisms and intersection with regular languages.