Test the divisibtlity of the following numbers by 11 :
(i) 4334 (ii) 83721 (iii) 66311 (iv) 137269 (v) 901351 (vi) 8790322

Answer

To test the divisibility of the given numbers by 11, we will follow a systematic approach. The rule for checking divisibility by 11 states that we should take the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions. If this difference is either 0 or a multiple of 11, then the number is divisible by 11.

Let's go through each number step by step:

### (i) 4334
1. Identify the digits in odd and even positions:
- Odd positions: 4 (1st), 3 (3rd) → Sum = 4 + 3 = 7
- Even positions: 3 (2nd), 4 (4th) → Sum = 3 + 4 = 7
2. Calculate the difference:
- Difference = 7 - 7 = 0
3. Since 0 is a multiple of 11, **4334 is divisible by 11**.

### (ii) 83721
1. Identify the digits in odd and even positions:
- Odd positions: 8 (1st), 7 (3rd), 1 (5th) → Sum = 8 + 7 + 1 = 16
- Even positions: 3 (2nd), 2 (4th) → Sum = 3 + 2 = 5
2. Calculate the difference:
- Difference = 16 - 5 = 11
3. Since 11 is a multiple of 11, **83721 is divisible by 11**.

### (iii) 66311
1. Identify the digits in odd and even positions:
- Odd positions: 6 (1st), 3 (3rd), 1 (5th) → Sum = 6 + 3 + 1 = 10
- Even positions: 6 (2nd), 1 (4th) → Sum = 6 + 1 = 7
2. Calculate the difference:
- Difference = 10 - 7 = 3
3. Since 3 is not a multiple of 11, **66311 is not divisible by 11**.

### (iv) 137269
1. Identify the digits in odd and even positions:
- Odd positions: 1 (1st), 7 (3rd), 6 (5th) → Sum = 1 + 7 + 6 = 14
- Even positions: 3 (2nd), 2 (4th), 9 (6th) → Sum = 3 + 2 + 9 = 14
2. Calculate the difference:
- Difference = 14 - 14 = 0
3. Since 0 is a multiple of 11, **137269 is divisible by 11**.

### (v) 901351
1. Identify the digits in odd and even positions:
- Odd positions: 9 (1st), 1 (3rd), 5 (5th) → Sum = 9 + 1 + 5 = 15
- Even positions: 0 (2nd), 3 (4th), 1 (6th) → Sum = 0 + 3 + 1 = 4
2. Calculate the difference:
- Difference = 15 - 4 = 11
3. Since 11 is a multiple of 11, **901351 is divisible by 11**.

### (vi) 8790322
1. Identify the digits in odd and even positions:
- Odd positions: 8 (1st), 9 (3rd), 3 (5th), 2 (7th) → Sum = 8 + 9 + 3 + 2 = 22
- Even positions: 7 (2nd), 0 (4th), 2 (6th) → Sum = 7 + 0 + 2 = 9
2. Calculate the difference:
- Difference = 22 - 9 = 13
3. Since 13 is not a multiple of 11, **8790322 is not divisible by 11**.

### Summary of Results:
- **Divisible by 11**: 4334, 83721, 137269, 901351
- **Not Divisible by 11**: 66311, 8790322