Which of the following are divisible by 2, 3 and 11 ?
A. 8448
B. 9812
C. 9126
D. 9636
Choose the correct option :

The correct option is A and D.

To determine which of the numbers are divisible by 2, 3, and 11, we must test each number against the rules of divisibility for 2, 3, and 11.

Divisibility Rules:
1. Divisibility by 2: The unit digit must be an even number (0, 2, 4, 6, 8).
2. Divisibility by 3: The sum of the digits must be divisible by 3.
3. Divisibility by 11: The difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11.

Checking Number A (8448):
- Last digit is 8 (even) ⇒ Divisible by 2.
- Sum of digits = $8+4+4+8=24$, which is divisible by 3 ⇒ Divisible by 3.
- Difference of alternate sums = $(8+4)-(4+8)=12-12=0$ ⇒ Divisible by 11.
Therefore, 8448 is divisible by 2, 3, and 11.

Checking Number B (9812):
- Last digit is 2 (even) ⇒ Divisible by 2.
- Sum of digits = $9+8+1+2=20$, which is not divisible by 3 ⇒ Not divisible by 3.
Therefore, 9812 is not divisible by all three numbers.

Checking Number C (9126):
- Last digit is 6 (even) ⇒ Divisible by 2.
- Sum of digits = $9+1+2+6=18$, which is divisible by 3 ⇒ Divisible by 3.
- Difference of alternate sums = $(9+2)-(1+6)=11-7=4$, which is not a multiple of 11 ⇒ Not divisible by 11.
Therefore, 9126 is not divisible by all three numbers.

Checking Number D (9636):
- Last digit is 6 (even) ⇒ Divisible by 2.
- Sum of digits = $9+6+3+6=24$, which is divisible by 3 ⇒ Divisible by 3.
- Difference of alternate sums = $(9+3)-(6+6)=12-12=0$ ⇒ Divisible by 11.
Therefore, 9636 is divisible by 2, 3, and 11.

Thus, numbers A and D satisfy all three conditions.