This question is on the GRE Computer Science test booklet (not homework). I tried applying closure properties of regular languages but no success.Suppose $L$ is a regular language over $\Sigma = \{0, 1\}$. Show that the language $\qquad L' = \{w \in L \mid |w| \in 2\mathbb{N}\}$ is also regular.What I find surprising is that the booklet mentions that the language $\{w \in L \mid |w| = 2^k, k \in \mathbb{N}\}$ is not necessarily a regular language. 
Proof that regular languages are closed against taking the even-length subset
formal languages;regular languages;closure properties