Test the divisibility of each of the following numbers by 3 :
(i) 83 (ii) 378 (iii) 474 (iv) 1693 (v) 20345 (vi) 67035 (vii) 591282 (viii) 903164 (ix) 100002

Answer

To test the divisibility of each of the given numbers by 3, we follow a simple rule: A number is divisible by 3 if the sum of its digits is divisible by 3. Let's apply this rule step by step for each number.

### Step-by-Step Solution:

1. **For 83:**
- Sum the digits: \(8 + 3 = 11\)
- Check if 11 is divisible by 3: \(11 \div 3 = 3\) remainder \(2\)
- **Conclusion:** 83 is **not divisible** by 3.

2. **For 378:**
- Sum the digits: \(3 + 7 + 8 = 18\)
- Check if 18 is divisible by 3: \(18 \div 3 = 6\) remainder \(0\)
- **Conclusion:** 378 is **divisible** by 3.

3. **For 474:**
- Sum the digits: \(4 + 7 + 4 = 15\)
- Check if 15 is divisible by 3: \(15 \div 3 = 5\) remainder \(0\)
- **Conclusion:** 474 is **divisible** by 3.

4. **For 1693:**
- Sum the digits: \(1 + 6 + 9 + 3 = 19\)
- Check if 19 is divisible by 3: \(19 \div 3 = 6\) remainder \(1\)
- **Conclusion:** 1693 is **not divisible** by 3.

5. **For 20345:**
- Sum the digits: \(2 + 0 + 3 + 4 + 5 = 14\)
- Check if 14 is divisible by 3: \(14 \div 3 = 4\) remainder \(2\)
- **Conclusion:** 20345 is **not divisible** by 3.

6. **For 67035:**
- Sum the digits: \(6 + 7 + 0 + 3 + 5 = 21\)
- Check if 21 is divisible by 3: \(21 \div 3 = 7\) remainder \(0\)
- **Conclusion:** 67035 is **divisible** by 3.

7. **For 591282:**
- Sum the digits: \(5 + 9 + 1 + 2 + 8 + 2 = 27\)
- Check if 27 is divisible by 3: \(27 \div 3 = 9\) remainder \(0\)
- **Conclusion:** 591282 is **divisible** by 3.

8. **For 903164:**
- Sum the digits: \(9 + 0 + 3 + 1 + 6 + 4 = 23\)
- Check if 23 is divisible by 3: \(23 \div 3 = 7\) remainder \(2\)
- **Conclusion:** 903164 is **not divisible** by 3.

9. **For 100002:**
- Sum the digits: \(1 + 0 + 0 + 0 + 0 + 2 = 3\)
- Check if 3 is divisible by 3: \(3 \div 3 = 1\) remainder \(0\)
- **Conclusion:** 100002 is **divisible** by 3.

### Final Results:
- 83: Not divisible by 3
- 378: Divisible by 3
- 474: Divisible by 3
- 1693: Not divisible by 3
- 20345: Not divisible by 3
- 67035: Divisible by 3
- 591282: Divisible by 3
- 903164: Not divisible by 3
- 100002: Divisible by 3