Question

By subtraction of successive odd numbers And whether the following numbers are perfect squares or not. 

(i) 55 

(ii) 90 

(iii) 121

Answer 1

(i) √55 

Step 1 → 55 – 1 = 54 (1st odd number be subtracted) 

Step 2 → 54 – 3 = 51 (2nd odd number be subtracted) 

Step 3 → 51 – 5 = 46 (3rd odd number be subtracted)

Step 4 → 46 – 7 = 39 (4th odd number be subtracted) 

Step 5 → 39 – 9 = 30 (5th odd number be subtracted) 

Step 6 → 30 – 11 = 19 (6th odd number be subtracted) 

Step 7 → 19 – 13 = 6 (7th odd number be subtracted)

∴ 55 is not a perfect square number.

(∵ difference of consecutive odd numbers is not equal to ‘0’)

ii) √90 

Step 1 → 90 – 1 =89 (1st odd number be subtracted) 

Step 2 → 89 – 3 = 86 (2nd odd number be subtracted) 

Step 3 → 86 – 5 = 81 (3rd odd number be subtracted) 

Step 4 → 81 – 7 = 74 (4th odd number be subtracted) 

Step 5 → 74 – 9 = 65 (5th odd number be subtracted) 

Step 6 → 65 – 11 = 54 (6th odd number be subtracted) 

Step 7 → 54 – 13 = 41 (7th odd number be subtracted) 

Step 8 → 41 – 15 = 26 (8th odd number be subtracted)

Step 9 → 26 – 17 = 9 (9th odd number be subtracted) 

∴ 90 is not a perfect square number. 

(∵ difference of consecutive odd numbers is not equal to ‘0’)

iii) √121 

Step 1 → 121 – 1 = 120 (1st odd number is subtracted)

Step 2 → 120 – 3 = 117 (2nd odd number is subtracted) 

Step 3 → 117 – 5 = 112 (3rd odd number is subtracted) 

Step 4 → 112 – 7 = 105 (4th odd number is subtracted) 

Step 5 → 105 – 9 = 96 (5th odd number is subtracted) 

Step 6 → 96 – 11 = 85 (6th odd number is subtracted) 

Step 7 → 85 – 13 = 72 (7th odd number is subtracted) 

Step 8 → 72 – 15 = 57 (8th odd number is subtracted)

Step 9 → 57 – 17 = 40 (9th odd number is subtracted) 

Step 10 → 40 – 19 = 21 (10th odd number is subtracted) 

Step 11 → 21 – 21 = 0 (11th odd number is subtracted)

∴ At the 11th step, the difference of consecutive odd numbers is ‘0’ 

121 is a perfect square number.

∴ √121 = \(\sqrt{11\,\times\,11}\) = 11 

(∵ It ends at 11th step)