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

(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