List of top Theory of Computation Questions on Regular expressions and finite automata

Let $ \Sigma = \{1,2,3,4\} $. For $ x \in \Sigma^* $, let $ {prod}(x) $ be the product of symbols in $ x $ modulo 7. We take $ {prod}(\epsilon) = 1 $, where $ \epsilon $ is the null string. For example, \[ {prod}(124) = (1 \times 2 \times 4) \mod 7 = 1. \] Define \[ L = \{ x \in \Sigma^* \mid {prod}(x) = 2 \}. \] The number of states in a minimum state DFA for $ L $ is ___________. (Answer in integer)

Which one of the following regular expressions correctly represents the language of the finite automaton given below?

Consider the following deterministic finite automaton (DFA). The number of strings of length 8 accepted by the above automaton is $\underline{\hspace{2cm}}$.

Which of the following regular expressions represent(s) the set of all binary numbers that are divisible by three? Assume that the string $\epsilon$ is divisible by three.

Consider the following language: \[ L = \{ w \in \{0,1\}^* \mid w \text{ ends with the substring } 011 \} \] Which one of the following deterministic finite automata accepts $L$?