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. 

1. 525 
2.  1750 
3.  252 
4.  1825
5.  6412

Answer 1

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

	22
2	\(\bar 5\bar 2 \bar 5\) \(-4\)
42	125 84
	41

The remainder is \(41\).
It represents that the square of \(22\) is less than \(525\).
Next number is \(23\) and \(23^2\)= \(529\)
Hence, number to be added to \(525 = 232- 525 = 529 - 525 = 4\)
The required perfect square is \(529\) and \(\sqrt{529} = 23\)

---

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

	41
4	\(\bar 1\bar 7 \bar 5\bar0\) \(-4\)
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 \(42^2\) = \(1764\)
Hence, number to be added to \(1750\) = 422 \(- 1750 = 1764 - 1750 = 14\)
The required perfect square is \(1764\) and \(\sqrt{1764} = 42\)

---

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

	15
1	\(\bar 2\bar 5 \bar 2\) \(-1\)
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 \(\sqrt{256}\) = \(16\)

---

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

	42
4	\(\bar 1\bar 8 \bar 2\bar5\) \(-16\)
82	225 164
	61

The remainder is \(61\). 
It represents that the square of \(42\)