To solve the problem, we need to check how many of the given numbers are divisible by 3 but not divisible by 9.

Steps:

1. Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
2. Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.

The numbers given are:
2133, 2343, 3474, 4131, 5286, 5340, 6336, 7347, 8115, 9276.

We will check each number.

Check each number:

1. 2133:
   Sum of digits = 2 + 1 + 3 + 3 = 9 (Divisible by 9) → Not included.

2. 2343:
   Sum of digits = 2 + 3 + 4 + 3 = 12 (Divisible by 3 but not 9) → Included.

3. 3474:
   Sum of digits = 3 + 4 + 7 + 4 = 18 (Divisible by 9) → Not included.

4. 4131:
   Sum of digits = 4 + 1 + 3 + 1 = 9 (Divisible by 9) → Not included.

5. 5286:
   Sum of digits = 5 + 2 + 8 + 6 = 21 (Divisible by 3 but not 9) → Included.

6. 5340:
   Sum of digits = 5 + 3 + 4 + 0 = 12 (Divisible by 3 but not 9) → Included.

7. 6336:
   Sum of digits = 6 + 3 + 3 + 6 = 18 (Divisible by 9) → Not included.

8. 7347:
   Sum of digits = 7 + 3 + 4 + 7 = 21 (Divisible by 3 but not 9) → Included.

9. 8115:
   Sum of digits = 8 + 1 + 1 + 5 = 15 (Divisible by 3 but not 9) → Included.

10. 9276:
    Sum of digits = 9 + 2 + 7 + 6 = 24 (Divisible by 3 but not 9) → Included.

Conclusion:

The numbers divisible by 3 but not by 9 are:
2343, 5286, 5340, 7347, 8115, 9276.

Total count = 6.

Answer: B: 6.

Do you want details on the calculations or further clarifications?

Here are some related questions to explore:

1. How do you check divisibility by other numbers, like 6 or 11?
2. What other divisibility rules are useful for larger numbers?
3. Can we apply divisibility tests to algebraic expressions?
4. How can we simplify divisibility testing for large sets of numbers?
5. Why does the sum of digits rule work for 3 and 9 but not for other numbers?

Tip: Knowing divisibility rules for small numbers (like 2, 3, 5, and 9) can save a lot of time in arithmetic and problem-solving.