GATE CSE 2025 Session 2 Q60: Minimal DFA for strings whose digit product mod 7 is 2

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, $prod(124) = (1 \times 2 \times 4) \bmod 7 = 1$.

Define $L = \{x \in \Sigma^* \mid prod(x) = 2\}$.

The number of states in a minimum state DFA for $L$ is ___________. (Answer in integer)