Test the divisibility of each of the following numbers by 7 :
(i) 693 (ii) 7896 (iii) 3467 (iv) 12873 (v) 65436 (vi) 54636 (vii) 98175 (viii) 88777

Answer

To test the divisibility of the given numbers by 7, we will use the divisibility rule of 7. The rule states that we can take the last digit of the number, double it, and subtract this from the rest of the number. If the result is either 0 or a multiple of 7, then the original number is divisible by 7.

Let's apply this rule step by step for each number:

### (i) 693
1. **Identify the last digit and the rest of the number**: Last digit = 3, Rest of the number = 69.
2. **Double the last digit**: 2 × 3 = 6.
3. **Subtract from the rest**: 69 - 6 = 63.
4. **Check if 63 is divisible by 7**: 63 ÷ 7 = 9 (which is a whole number).
5. **Conclusion**: 693 is divisible by 7.

### (ii) 7896
1. **Identify the last digit and the rest of the number**: Last digit = 6, Rest of the number = 789.
2. **Double the last digit**: 2 × 6 = 12.
3. **Subtract from the rest**: 789 - 12 = 777.
4. **Check if 777 is divisible by 7**: 777 ÷ 7 = 111 (which is a whole number).
5. **Conclusion**: 7896 is divisible by 7.

### (iii) 3467
1. **Identify the last digit and the rest of the number**: Last digit = 7, Rest of the number = 346.
2. **Double the last digit**: 2 × 7 = 14.
3. **Subtract from the rest**: 346 - 14 = 332.
4. **Check if 332 is divisible by 7**: 332 ÷ 7 = 47.42857 (not a whole number).
5. **Conclusion**: 3467 is not divisible by 7.

### (iv) 12873
1. **Identify the last digit and the rest of the number**: Last digit = 3, Rest of the number = 1287.
2. **Double the last digit**: 2 × 3 = 6.
3. **Subtract from the rest**: 1287 - 6 = 1281.
4. **Check if 1281 is divisible by 7**: 1281 ÷ 7 = 183 (which is a whole number).
5. **Conclusion**: 12873 is divisible by 7.

### (v) 65436
1. **Identify the last digit and the rest of the number**: Last digit = 6, Rest of the number = 6543.
2. **Double the last digit**: 2 × 6 = 12.
3. **Subtract from the rest**: 6543 - 12 = 6531.
4. **Check if 6531 is divisible by 7**: 6531 ÷ 7 = 933 (which is a whole number).
5. **Conclusion**: 65436 is divisible by 7.

### (vi) 54636
1. **Identify the last digit and the rest of the number**: Last digit = 6, Rest of the number = 5463.
2. **Double the last digit**: 2 × 6 = 12.
3. **Subtract from the rest**: 5463 - 12 = 5451.
4. **Check if 5451 is divisible by 7**: 5451 ÷ 7 = 778.7142857 (not a whole number).
5. **Conclusion**: 54636 is not divisible by 7.

### (vii) 98175
1. **Identify the last digit and the rest of the number**: Last digit = 5, Rest of the number = 9817.
2. **Double the last digit**: 2 × 5 = 10.
3. **Subtract from the rest**: 9817 - 10 = 9807.
4. **Check if 9807 is divisible by 7**: 9807 ÷ 7 = 1401 (which is a whole number).
5. **Conclusion**: 98175 is divisible by 7.

### (viii) 88777
1. **Identify the last digit and the rest of the number**: Last digit = 7, Rest of the number = 8877.
2. **Double the last digit**: 2 × 7 = 14.
3. **Subtract from the rest**: 8877 - 14 = 8863.
4. **Check if 8863 is divisible by 7**: 8863 ÷ 7 = 1266.142857 (not a whole number).
5. **Conclusion**: 88777 is not divisible by 7.

### Summary of Results:
- 693: Divisible by 7
- 7896: Divisible by 7
- 3467: Not divisible by 7
- 12873: Divisible by 7
- 65436: Divisible by 7
- 54636: Not divisible by 7
- 98175: Divisible by 7
- 88777: Not divisible by 7