Is set of words a regular language?

Question

Is the set of all words from a regular language A, where every word has a length divided by 2, a regular language?

I was trying to find some counterexample, but without results.

Answer 1

Let $L$ be a regular language on the alphabet $A$. Then the language you consider is the intersection of $L$ with the regular language $(A^2)^*$ of all words of even length. Since the intersection of two regular languages is regular, your language is regular.