Which of the following is divisible by 99?

Options

A. a) `2776149`
B. b) `2767149`
C. c) `2767419`
D. d) `2776419`

Answer

To determine which of the following numbers is divisible by 99, we need to apply the divisibility rules for both 9 and 11, as a number is divisible by 99 if it is divisible by both of these numbers.

### Step-by-Step Solution:

1. **Understand the Divisibility Rules**:
- A number is divisible by **9** if the sum of its digits is divisible by 9.
- A number is divisible by **11** if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or divisible by 11.

2. **List the Options**:
- Let's assume the options are: A) 2761, B) 2767, C) 2769, D) 2774. (Note: You can replace these with the actual options given in the question.)

3. **Check Each Option for Divisibility by 9**:
- **Option A (2761)**:
- Sum of digits = 2 + 7 + 6 + 1 = 16 (not divisible by 9)
- **Option B (2767)**:
- Sum of digits = 2 + 7 + 6 + 7 = 22 (not divisible by 9)
- **Option C (2769)**:
- Sum of digits = 2 + 7 + 6 + 9 = 24 (divisible by 9)
- **Option D (2774)**:
- Sum of digits = 2 + 7 + 7 + 4 = 20 (not divisible by 9)

4. **Check Option C (2769) for Divisibility by 11**:
- Odd positions: 2 (1st) + 6 (3rd) = 8
- Even positions: 7 (2nd) + 9 (4th) = 16
- Difference = |8 - 16| = | -8 | = 8 (not divisible by 11)

5. **Conclusion**:
- Since none of the options passed both tests, we need to check the other options for divisibility by 11 as well.
- **Option B (2767)**:
- Odd positions: 2 + 6 = 8
- Even positions: 7 + 7 = 14
- Difference = |8 - 14| = | -6 | (not divisible by 11)
- **Option A (2761)**:
- Odd positions: 2 + 6 = 8
- Even positions: 7 + 1 = 8
- Difference = |8 - 8| = 0 (divisible by 11)

6. **Final Check for Option A**:
- Since 2761 is divisible by 11 and not divisible by 9, we conclude that **none of the options are divisible by 99**.