Number of states in minimal FA for L = {(a b) × | number of a's = (0 mod 4) and (0 mod 8)}

Introduction
The given language L consists of all strings of the form (a^m b^n) where the number of a's is divisible by both 4 and 8. We need to find the number of states in the minimal finite automaton (FA) for this language.

Approach
To construct the minimal FA, we need to consider the possible remainders when the number of a's is divided by 4 and 8. We can represent these remainders using two states. Let's analyze the steps to construct the minimal FA for L.

Step 1: Remainder 0 (mod 4) and 0 (mod 8)
The initial state of the FA will represent the remainder 0 (mod 4) and 0 (mod 8) since the empty string is also in the language. Let's denote this state as A.

Step 2: Remainder 0 (mod 4) and 1 (mod 8)
If we read a 'b' in state A, the number of b's increases by 1, but the number of a's remains the same. This leads to a remainder of 0 (mod 4) and 1 (mod 8). We can transition to a new state, B, to represent this remainder.

Step 3: Remainder 1 (mod 4) and 2 (mod 8)
If we read another 'b' in state B, the number of b's increases by 1 again, but the number of a's remains the same. This results in a remainder of 1 (mod 4) and 2 (mod 8). We can transition to a new state, C, to represent this remainder.

Step 4: Remainder 2 (mod 4) and 4 (mod 8)
If we read a 'b' in state C, the number of b's increases by 1 once more, but the number of a's remains the same. This gives us a remainder of 2 (mod 4) and 4 (mod 8). We can transition to a new state, D, to represent this remainder.

Step 5: Remainder 3 (mod 4) and 0 (mod 8)
If we read a 'b' in state D, the number of b's increases by 1 again, but the number of a's remains the same. This leads to a remainder of 3 (mod 4) and 0 (mod 8). We can transition back to state A (initial state) to represent this remainder.

Step 6: Remainder 0 (mod 4) and 3 (mod 8)
If we read an 'a' in state A, the number of a's increases by 1, but the number of b's remains the same. This results in a remainder of 0 (mod 4) and 3 (mod 8). We can transition to a new state, E, to represent this remainder.

Step 7: Remainder 1