Problem $18.$ Prove that for any infinite regular language $L,$ there exist two infinite regular languages $L_1,L_2$ such that $L=L_1{L}_2$ and $L_1{L}_2=\frac{O}{?}.$ Here are some hints.

Let $D$ be any DFA and $q$ be any one of its states. Prove that the language

$L_q=\{x|xinL(D)$ and the run $ofDonx$ visits $qan$ odd number $of$ times $\}$

is regular.

$2.$ Recall the proof of the pumping lemma.