Question

Find the square roots of 100 and 169 by the method of repeated subtraction.

Answer 1

We know that the sum of the first n odd natural numbers is n².

Consider `sqrt100`

1. 100 − 1 = 99
2. 99 − 3 = 96
3. 96 − 5 = 91
4. 91 − 7 = 84
5. 84 − 9 = 75
6. 75 − 11= 64
7. 64 − 13 = 51
8. 51 − 15 = 36
9. 36 − 17 = 19
10. 19 − 19 = 0

We have subtracted successive odd numbers starting from 1 to 100, and obtained 0 at 10^th step.

Therefore, `sqrt100 = 10`

The square root of 169 can be obtained by the method of repeated subtraction as follows.

1. 169 − 1 = 168
2. 168 − 3 = 165
3. 165 − 5 = 160
4. 160 − 7 = 153
5. 153 − 9 = 144
6. 144 − 11 = 133
7. 133 − 13 = 120
8. 120 − 15 = 105
9. 105 − 17 = 88
10. 88 − 19 = 69
11. 69 − 21 = 48
12. 48 − 23 = 25
13. 25 − 25 = 0

We have subtracted successive odd numbers starting from 1 to 169, and obtained 0 at 13^th step.

Therefore, `sqrt169 = 13`