Question

Use the method of successive subtraction of odd numbers (starting from 1) to examine the following numbers and find if they are perfect squares or not. For perfect square numbers that you find, identify the square root.

(i) 144

(ii) 256

(iii) 360

Answer 1

(i) 144

Step 1: 144 – 1 = 143
Step 2: 143 – 3 = 140
Step 3: 140 – 5 = 135
Step 4: 135 – 7 = 128
Step 5: 128 – 9 = 119
Step 6: 119 – 11 = 108
Step 7: 108 – 13 = 95
Step 8: 95 – 15 = 80
Step 9: 80 – 17 = 63
Step 10: 63 – 19 = 44
Step 11: 44 – 21 = 23
Step 12: 23 – 23 = 0.

We get 0 in the 12th step. 

∴ 144 is a perfect square and 

∴ √144 = 12.

(ii) 256

Step 1: 256 – 1 = 255
Step 2: 255 – 3 = 252
Step 3: 252 – 5 = 247
Step 4: 247 – 7 = 240
Step 5: 240 – 9 = 231
Step 6: 231 – 11 = 220
Step 7: 220 – 13 = 207
Step 8: 207 – 15 = 192
Step 9: 192 – 17 = 175
Step 10: 175 – 19 = 156
Step 11: 156 – 21 = 135
Step 12: 135 – 23 = 112
Step 13: 112 – 25 = 87
Step 14: 87 – 27 = 60
Step 15: 60 – 29 = 31
Step 16: 31 – 31 = 0

We have subtracted odd numbers starting from 1 repeatedly from 256. We get 0 in the 1 step.

∴ 256 is a perfect square and √256 = 16.

(iii) 360

Step 1: 360 – 1 = 359
Step 2: 359 – 3 = 356
Step 3: 356 – 5 = 351
Step 4: 351 – 7 = 344
Step 5: 344 – 9 = 335
Step 6: 335 – 11 = 324
Step 7: 324 – 13 = 311
Step 8: 311 – 15 = 296
Step 9: 296 – 17 = 279
Step 10: 279 – 19 = 260
Step 11: 260 – 21 = 239
Step 12: 239 – 23 = 216
Step 13: 216 – 25 = 191
Step 14: 191 – 27 = 164
Step 15: 164 – 29 = 135
Step 16: 135 – 31 = 104
Step 17: 104 – 33 = 71
Step 18: 71 – 35 = 36
Step 19: 36 – 37 = -1

We have subtracted successive odd numbers starting from 1, repeatedly from 360, we didn’t get zero.

∴ The given number is not perfect square.