To determine the divisibility of the given numbers by 11, we will use the divisibility rule for 11. According to this rule, we need to find the sum of the digits in the odd positions and the sum of the digits in the even positions. Then, we calculate the difference between these two sums. If the difference is divisible by 11 (including zero), then the original number is also divisible by 11.
Let's solve each part step by step:
### (i) 22222
1. **Identify the digits in odd and even positions:**
- Odd positions: 2 (1st), 2 (3rd), 2 (5th) → Sum = 2 + 2 + 2 = 6
- Even positions: 2 (2nd), 2 (4th) → Sum = 2 + 2 = 4
2. **Calculate the difference:**
- Difference = Sum of odd positions - Sum of even positions = 6 - 4 = 2
3. **Check divisibility:**
- 2 is not divisible by 11.
- **Conclusion:** 22222 is **not divisible by 11**.
### (ii) 444444
1. **Identify the digits in odd and even positions:**
- Odd positions: 4 (1st), 4 (3rd), 4 (5th) → Sum = 4 + 4 + 4 = 12
- Even positions: 4 (2nd), 4 (4th), 4 (6th) → Sum = 4 + 4 + 4 = 12
2. **Calculate the difference:**
- Difference = 12 - 12 = 0
3. **Check divisibility:**
- 0 is divisible by 11.
- **Conclusion:** 444444 is **divisible by 11**.
### (iii) 379654
1. **Identify the digits in odd and even positions:**
- Odd positions: 3 (1st), 9 (3rd), 5 (5th) → Sum = 3 + 9 + 5 = 17
- Even positions: 7 (2nd), 6 (4th), 4 (6th) → Sum = 7 + 6 + 4 = 17
2. **Calculate the difference:**
- Difference = 17 - 17 = 0
3. **Check divisibility:**
- 0 is divisible by 11.
- **Conclusion:** 379654 is **divisible by 11**.
### (iv) 1057982
1. **Identify the digits in odd and even positions:**
- Odd positions: 1 (1st), 5 (3rd), 9 (5th), 2 (7th) → Sum = 1 + 5 + 9 + 2 = 17
- Even positions: 0 (2nd), 7 (4th), 8 (6th) → Sum = 0 + 7 + 8 = 15
2. **Calculate the difference:**
- Difference = 17 - 15 = 2
3. **Check divisibility:**
- 2 is not divisible by 11.
- **Conclusion:** 1057982 is **not divisible by 11**.
### (v) 6543207
1. **Identify the digits in odd and even positions:**
- Odd positions: 6 (1st), 4 (3rd), 2 (5th), 7 (7th) → Sum = 6 + 4 + 2 + 7 = 19
- Even positions: 5 (2nd), 3 (4th), 0 (6th) → Sum = 5 + 3 + 0 = 8
2. **Calculate the difference:**
- Difference = 19 - 8 = 11
3. **Check divisibility:**
- 11 is divisible by 11.
- **Conclusion:** 6543207 is **divisible by 11**.
### (vi) 818532
1. **Identify the digits in odd and even positions:**
- Odd positions: 8 (1st), 8 (3rd), 3 (5th) → Sum = 8 + 8 + 3 = 19
- Even positions: 1 (2nd), 5 (4th), 2 (6th) → Sum = 1 + 5 + 2 = 8
2. **Calculate the difference:**
- Difference = 19 - 8 = 11
3. **Check divisibility:**
- 11 is divisible by 11.
- **Conclusion:** 818532 is **divisible by 11**.
### (vii) 900163
1. **Identify the digits in odd and even positions:**
- Odd positions: 9 (1st), 0 (3rd), 6 (5th) → Sum = 9 + 0 + 6 = 15
- Even positions: 0 (2nd), 1 (4th), 3 (6th) → Sum = 0 + 1 + 3 = 4
2. **Calculate the difference:**
- Difference = 15 - 4 = 11
3. **Check divisibility:**
- 11 is divisible by 11.
- **Conclusion:** 900163 is **divisible by 11**.
### (viii) 7531622
1. **Identify the digits in odd and even positions:**
- Odd positions: 7 (1st), 3 (3rd), 1 (5th), 2 (7th) → Sum = 7 + 3 + 1 + 2 = 13
- Even positions: 5 (2nd), 6 (4th), 2 (6th) → Sum = 5 + 6 + 2 = 13
2. **Calculate the difference:**
- Difference = 13 - 13 = 0
3. **Check divisibility:**
- 0 is divisible by 11.
- **Conclusion:** 7531622 is **divisible by 11**.
### Summary of Results:
- 22222: Not divisible by 11
- 444444: Divisible by 11
- 379654: Divisible by 11
- 1057982: Not divisible by 11
- 6543207: Divisible by 11
- 818532: Divisible by 11
- 900163: Divisible by 11
- 7531622: Divisible by 11
