Question

Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

(i) 525 (ii) 1750

(iii) 252 (iv) 1825

(v) 6412

Answer 1

(i) The square root of 525 can be calculated by long division method as follows.

	22
2
42	125 84
	41

The remainder is 41.

It represents that the square of 22 is less than 525.

Next number is 23 and 23² = 529

Hence, number to be added to 525 = 23² − 525 = 529 − 525 = 4

The required perfect square is 529 and

(ii) The square root of 1750 can be calculated by long division method as follows.

	41
4
81	150 81
	69

The remainder is 69.

It represents that the square of 41 is less than 1750.

The next number is 42 and 42² = 1764

Hence, number to be added to 1750 = 42² − 1750 = 1764 − 1750 = 14

The required perfect square is 1764 and

(iii) The square root of 252 can be calculated by long division method as follows.

	15
1
25	152 125
	27

The remainder is 27. It represents that the square of 15 is less than 252.

The next number is 16 and 16² = 256

Hence, number to be added to 252 = 16² − 252 = 256 − 252 = 4

The required perfect square is 256 and

(iv) The square root of 1825 can be calculated by long division method as follows.

	42
4
82	225 164
	61

The remainder is 61. It represents that the square of 42 is less than 1825.

The next number is 43 and 43² = 1849

Hence, number to be added to 1825 = 43² − 1825 = 1849 − 1825 = 24

The required perfect square is 1849 and

(v) The square root of 6412 can be calculated by long division method as follows.

	80
8
160	012 0
	12

The remainder is 12.

It represents that the square of 80 is less than 6412.

The next number is 81 and 81² = 6561

Hence, number to be added to 6412 = 81² − 6412 = 6561 − 6412 = 149

The required perfect square is 6561 and