Question

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)

Hint:

For problems involving modular arithmetic in a DFA, consider the number of possible distinct residue classes as the number of states required.

Answer 1

The function \( {prod}(x) \) maps strings over \( \Sigma \) to values in the set \( \{0,1,2,3,4,5,6\} \) modulo 7. Since this function tracks the product of elements modulo 7, it defines a residue class system of at most 7 possible values. 
Step 1: Compute the transition function for modulo 7 residues Each input character \( c \in \Sigma \) modifies the current residue \( r \) via multiplication \( (r \times c) \mod 7 \). Since all values map to one of 7 possible residues, the DFA must have at most 7 states. 
Step 2: Identify the minimum number of states - The DFA needs one state for each residue \( 0,1,2,3,4,5,6 \) modulo 7. - The accepting state is the one corresponding to residue 2. - Since multiplication modulo 7 never produces 0 for any sequence of nonzero values, state 0 is unreachable. Thus, the minimal DFA requires 6 states (corresponding to residues \( 1,2,3,4,5,6 \)).