How to demonstrate that a language is regular for k?
Generalizing the other answer, here is a complete classification of regular language of this sort.


  The language $\{ x \in \Sigma^* : |x| \in S \}$ is regular iff there exist $n \geq 0$ and $k \geq 1$ and sets $A \subseteq \{0,\ldots,n-1\}$, $B \subseteq \{0,\ldots,k-1\}$ such that
  $$ S = A \cup \{ n+ mk + r : m \geq 0, r \in B\}.$$


One direction follows from a direct construction (either a DFA or a regular expression), and the other from the pumping lemma (exercise) or Parikh's theorem.