Definition of the language $$L$$ with alphabet $$\left\{ a \right\}$$ is given as following. $$L = \left\{ {{a^{nk}}} \right.\left| {k > 0,\,n} \right.$$ is a positive integer constant$$\left. \, \right\}$$

What is the minimum number of states needed in a $$DFA$$ to recognize $$L$$?

Let $$L = \left\{ {w \in {{\left( {0 + 1} \right)}^ * }\left| {\,w} \right.} \right.$$ has even number of $$\,\left. {1's} \right\},$$ i.e $$L$$ is the set of all bit strings with even number of $$1's.$$ which one of rhe regular expression below represents $$L.$$

Let $$w$$ be any string of length $$n$$ in $${\left\{ {0,1} \right\}^ * }$$. Let $$L$$ be the set of all substrings of $$w.$$ What is the minimum number of states in a non-deterministic finite automation that accepts $$L$$?

$$L = {L_1} \cap {L_2}$$ where $${L_1}$$ and $${L_2}$$ are languages defined as follows.
$${L_1} = \left\{ {{a^m}{b^m}\,c\,{a^n}{b^n}\left| {m,n \ge 0} \right.} \right\}$$
$${L_2} = \left\{ {{a^i}{b^i}{c^k}\left| {i,j,k \ge 0} \right.} \right\}$$ Then $$L$$ is