Question

Find the square root of 121 by repeated subtraction.

Answer 1

Square numbers have a pattern where each square number can be represented as a sum of consecutive odd numbers.

Let us start subtracting odd numbers in order from 1, 3, . . .

121 $-$ 1 = 120

120 $-$ 3 = 117

117 $-$ 5 = 112

112 $-$ 7 = 105

105 $-$ 9 = 96

96 $-$ 11 = 85

85 $-$ 13 = 72

72 $-$ 15 = 57

57 $-$ 17 = 40

40 $-$ 19 = 21

21 $-$ 21 = 0

Let us count the number of odd numbers subtracted. We see that we have subtracted 11 odd numbers from 121.

So, $\sqrt{121}$ = 11