Explanation:

To recognize an octal number divisible by 3, we can use a finite automaton. A finite automaton consists of states and transitions between states based on input symbols.

In this case, the input symbols are the digits 0, 1, 2, 3, 4, 5, 6, and 7 (since the number is octal). We need to design a finite automaton that accepts the octal numbers divisible by 3.

1. Modulo 3 Property:
To determine if a number is divisible by 3, we can use the modulo 3 property. If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3.

2. Designing the Finite Automaton:
To recognize an octal number divisible by 3, we can design a finite automaton with 3 states. Let's call them state 0, state 1, and state 2.

3. Transition Table:
The transition table for the finite automaton will have 3 rows (one for each state) and 8 columns (one for each input symbol).

State 0:
- Transition on input 0: Goes to state 0
- Transition on input 1: Goes to state 1
- Transition on input 2: Goes to state 2
- Transition on input 3: Goes to state 0
- Transition on input 4: Goes to state 1
- Transition on input 5: Goes to state 2
- Transition on input 6: Goes to state 0
- Transition on input 7: Goes to state 1

State 1:
- Transition on input 0: Goes to state 1
- Transition on input 1: Goes to state 2
- Transition on input 2: Goes to state 0
- Transition on input 3: Goes to state 1
- Transition on input 4: Goes to state 2
- Transition on input 5: Goes to state 0
- Transition on input 6: Goes to state 1
- Transition on input 7: Goes to state 2

State 2:
- Transition on input 0: Goes to state 2
- Transition on input 1: Goes to state 0
- Transition on input 2: Goes to state 1
- Transition on input 3: Goes to state 2
- Transition on input 4: Goes to state 0
- Transition on input 5: Goes to state 1
- Transition on input 6: Goes to state 2
- Transition on input 7: Goes to state 0

4. Accepting State:
State 0 is the accepting state, meaning that if the finite automaton ends up in state 0 after reading all the input symbols, the input octal number is divisible by 3.

5. Example:
Let's take an example with the octal number 56:
- Start at state 0
- Transition on input 5: Goes to state 2
- Transition on input 6: Goes to state 0
- End at state 0 (accepting state)

Since the finite automaton ends up in state 0