Explanation:

To construct a DFA that accepts all strings over {a, b} where the number of a's mod 3 = 2 and the number of b's are odd, we need to consider the possible combinations of a's mod 3 and odd b's.

States:
Let's consider the possible combinations of a's mod 3:
- If the number of a's mod 3 = 0, we can represent it with state A.
- If the number of a's mod 3 = 1, we can represent it with state B.
- If the number of a's mod 3 = 2, we can represent it with state C.

Transitions:
From each state, we need to consider the possible transitions based on the next input symbol:
1. If the next input symbol is 'a', the number of a's mod 3 will increment by 1. So, the transitions will be:
- From state A, if the next input symbol is 'a', we will transition to state B.
- From state B, if the next input symbol is 'a', we will transition to state C.
- From state C, if the next input symbol is 'a', we will transition back to state A.
2. If the next input symbol is 'b', the number of b's will increment by 1. So, the transitions will be:
- From state A, if the next input symbol is 'b', we will stay in state A.
- From state B, if the next input symbol is 'b', we will transition to state B.
- From state C, if the next input symbol is 'b', we will stay in state C.

Accepting States:
We need to define the accepting states based on the conditions given:
- The number of a's mod 3 = 2 (state C).
- The number of b's is odd.

DFA Diagram:
Based on the above transitions and accepting states, we can construct the DFA diagram as follows:

```
a
------------
| |
v |
---> A ---> B --|
| ^ | |
| | b |
| | | |
-- C < />
```

Minimum Number of States:
To find the minimum number of states required for this DFA, we can use the concept of Myhill-Nerode equivalence. In this case, there are 3 distinct combinations of a's mod 3 (0, 1, 2) and 2 distinct possibilities for the number of b's (odd, even). So, the total number of distinct combinations is 3 * 2 = 6. Therefore, the minimum number of states required for this DFA is 6.

Hence, the correct answer is option C) 6.