Prove $ L = \{ww^{R} \in \{a, b\}^{*} : |w|_{a} \equiv |w|_{b} \equiv 0$ $ (mod$ $13) \} $ is regular or context-free or neither
Let 
$$ P = \{ww^{R} : w\in \{a, b\}^{*} \} $$
$$R_a=\{u\in \{a, b\}^{*} : |u|_{a}\equiv 0\ (\text{mod}13)  \} $$
$$R_b=\{u\in \{a, b\}^{*} : |u|_{b}\equiv 0\ (\text{mod}13)  \} $$

We know $P$ is context-free. 

We know $R_a$ is regular and $R_b$ is regular. So $R_a\cap R_b$ is regular.

So, $L=P\cap (R_a\cap R_b)$ is context-free.



Exercise 1. (One minute or less) Verify that $L=P\cap (R_a\cap R_b)$.

Exercise 2. Show that $R_a$ is regular.