Given an arbitrary non-deterministic finite automaton $$(NFA)$$ with $$N$$ states, the maximum number of states in an equivalent minimized $$DFA$$ is at least

Consider a $$DFA$$ over $$\sum { = \left\{ {a,\,\,b} \right\}} $$ accepting all strings which have number of $$a'$$s divisible by $$6$$ and number of $$b'$$s divisible by $$8$$. What is the minimum number of states that the $$DFA$$ will have?

Consider the following languages:
$${L_1} = \left\{ {w\,w\left| {w \in {{\left\{ {a,\,b} \right\}}^ * }} \right.} \right\}$$
$${L_2} = \left\{ {w\,{w^R}\left| {w \in {{\left\{ {a,\,b} \right\}}^ * },} \right.{w^R}\,\,} \right.$$ is the reverse of $$\left. w \right\}$$
$${L_3} = \left\{ {{0^{2i}}\left| {i\,\,} \right.} \right.$$ is an integer$$\left. \, \right\}$$
$${L_4} = \left\{ {{0^{{i^2}}}\left| {i\,\,} \right.} \right.$$ is an integer$$\left. \, \right\}$$

Which of the languages are regular?

Which of the following statement is true?