Test the divisibility of the following numbers by 9 :
(i) 2358 (ii) 3333 (iii) 98712 (iv) 257106 (v) 647514 (vi) 326999

Answer

To determine if the given numbers are divisible by 9, we will use the rule that states: A number is divisible by 9 if the sum of its digits is divisible by 9. Let's apply this rule step by step for each number.

### Step-by-Step Solution:

**(i) For the number 2358:**
1. Add the digits: \(2 + 3 + 5 + 8\)
2. Calculate:
- \(2 + 3 = 5\)
- \(5 + 5 = 10\)
- \(10 + 8 = 18\)
3. Check divisibility: \(18 \div 9 = 2\) (exact division)
4. Conclusion: **2358 is divisible by 9.**

**(ii) For the number 3333:**
1. Add the digits: \(3 + 3 + 3 + 3\)
2. Calculate:
- \(3 + 3 = 6\)
- \(6 + 3 = 9\)
- \(9 + 3 = 12\)
3. Check divisibility: \(12 \div 9 = 1.33\) (not an exact division)
4. Conclusion: **3333 is not divisible by 9.**

**(iii) For the number 98712:**
1. Add the digits: \(9 + 8 + 7 + 1 + 2\)
2. Calculate:
- \(9 + 8 = 17\)
- \(17 + 7 = 24\)
- \(24 + 1 = 25\)
- \(25 + 2 = 27\)
3. Check divisibility: \(27 \div 9 = 3\) (exact division)
4. Conclusion: **98712 is divisible by 9.**

**(iv) For the number 257106:**
1. Add the digits: \(2 + 5 + 7 + 1 + 0 + 6\)
2. Calculate:
- \(2 + 5 = 7\)
- \(7 + 7 = 14\)
- \(14 + 1 = 15\)
- \(15 + 0 = 15\)
- \(15 + 6 = 21\)
3. Check divisibility: \(21 \div 9 = 2.33\) (not an exact division)
4. Conclusion: **257106 is not divisible by 9.**

**(v) For the number 647514:**
1. Add the digits: \(6 + 4 + 7 + 5 + 1 + 4\)
2. Calculate:
- \(6 + 4 = 10\)
- \(10 + 7 = 17\)
- \(17 + 5 = 22\)
- \(22 + 1 = 23\)
- \(23 + 4 = 27\)
3. Check divisibility: \(27 \div 9 = 3\) (exact division)
4. Conclusion: **647514 is divisible by 9.**

**(vi) For the number 326999:**
1. Add the digits: \(3 + 2 + 6 + 9 + 9 + 9\)
2. Calculate:
- \(3 + 2 = 5\)
- \(5 + 6 = 11\)
- \(11 + 9 = 20\)
- \(20 + 9 = 29\)
- \(29 + 9 = 38\)
3. Check divisibility: \(38 \div 9 = 4.22\) (not an exact division)
4. Conclusion: **326999 is not divisible by 9.**

### Final Results:
- **2358**: Divisible by 9
- **3333**: Not divisible by 9
- **98712**: Divisible by 9
- **257106**: Not divisible by 9
- **647514**: Divisible by 9
- **326999**: Not divisible by 9