If the 5-digit number $535ab$ is divisible by $3$, $7$, and $11$, then what is the value of $a^2 - b^2 + ab$?

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**Solution:**

1. The number is $535ab$, where $a$ and $b$ are digits.

2. The number is divisible by $3$, $7$, and $11$. Since these are prime factors, the number must be divisible by their least common multiple (LCM).

3. Calculate LCM of $3$, $7$, and $11$:
   - $3$, $7$, and $11$ are all prime and distinct.
   - So, $	ext{LCM} = 3 	imes 7 	imes 11 = 231$.

4. Therefore, $535ab$ must be divisible by $231$.

5. Express the number $535ab$ as:
   $$535ab = 53500 + 10a + b$$

6. We want to find digits $a$ and $b$ such that:
   $$53500 + 10a + b \equiv 0 \pmod{231}$$

7. Find $53500 \bmod 231$:
   - Divide $53500$ by $231$:
     - $231 	imes 231 = 53361$ (since $231^2 = 53361$)
     - $53500 - 53361 = 139$
   - So, $53500 \equiv 139 \pmod{231}$

8. Therefore:
   $$139 + 10a + b \equiv 0 \pmod{231}$$
   $$10a + b \equiv -139 \equiv 231 - 139 = 92 \pmod{231}$$

9. Since $a$ and $b$ are digits (0 to 9), $10a + b$ is between 0 and 99.

10. So, 
    $$10a + b = 92$$

11. Possible pairs $(a,b)$ satisfying $10a + b = 92$ are:
    - $a = 9$, $b = 2$

12. Check divisibility by $3$:
    - Sum of digits: $5 + 3 + 5 + 9 + 2 = 24$
    - $24$ is divisible by $3$, so condition holds.

13. Check divisibility by $7$ and $11$:
    - Since the number is divisible by $231$, it is divisible by $7$ and $11$ automatically.

14. Now calculate:
    $$a^2 - b^2 + ab = 9^2 - 2^2 + 9 	imes 2 = 81 - 4 + 18 = 95$$

15. But $95$ is option (a), however, let's verify the divisibility carefully.

**Answer:** $95$

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