Solution:
An octal number is divisible by 3 if and only if the sum of its digits is divisible by 3.

To recognize an octal number divisible by 3, we need to design an automaton that accepts the language of all octal numbers whose sum of digits is divisible by 3.

Let us design such an automaton using the state transition diagram.

State transition diagram:
The state transition diagram of the automaton for recognizing an octal number divisible by 3 is as follows:

[image]

Explanation:
The automaton has three states, namely, state 0, state 1, and state 2. Each state corresponds to the remainder of the sum of digits of the input octal number modulo 3.

The initial state is state 0, which corresponds to the remainder 0. The final state is also state 0, because if the sum of digits of the input octal number is divisible by 3, then its remainder modulo 3 is also 0.

The transition function is defined as follows:
- If the current state is state 0 and the input symbol is 0, the next state is state 0.
- If the current state is state 0 and the input symbol is 1, the next state is state 1.
- If the current state is state 0 and the input symbol is 2, the next state is state 2.
- If the current state is state 1 and the input symbol is 0, the next state is state 2.
- If the current state is state 1 and the input symbol is 1, the next state is state 0.
- If the current state is state 1 and the input symbol is 2, the next state is state 1.
- If the current state is state 2 and the input symbol is 0, the next state is state 1.
- If the current state is state 2 and the input symbol is 1, the next state is state 2.
- If the current state is state 2 and the input symbol is 2, the next state is state 0.

The automaton accepts an input octal number if and only if its final state is state 0.

Conclusion:
Hence, the minimum number of states required to recognize an octal number divisible by 3 is 3. Therefore, option 'B' is the correct answer.

Solution:

To solve this problem, we will construct a DFA (Deterministic Finite Automaton) that accepts all octal numbers divisible by 3.

Steps to Construct DFA:

Step 1: First, we will create a DFA that accepts all octal numbers.

- The DFA will have eight states, one for each digit from 0 to 7.
- The initial state will be the state for digit 0, and all other states will be non-final states.
- For each digit, there will be eight outgoing edges, one for each possible next digit.

Step 2: Next, we will modify the DFA to accept only octal numbers divisible by 3.

- We know that an octal number is divisible by 3 if and only if the sum of its digits is divisible by 3.
- Therefore, we will add three new states to the DFA, one for each possible remainder when the sum of the digits is divided by 3.
- We will make the initial state the state for digit 0 and remainders 0.
- We will make the state for digit 0 and remainders 0 a final state.
- For each digit, we will update the outgoing edges to transition to the appropriate remainder state based on the current remainder and the value of the digit.

Step 3: Finally, we will simplify the DFA by merging equivalent states.

- We can merge any two states that have the same outgoing edges and the same finality.
- Using this method, we can reduce the DFA to only three states, which are the minimum number of states required to recognize an octal number divisible by 3.

Hence, the correct answer is option 'B', i.e., 3 states are required to recognize an octal number divisible by 3.