Question

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

1. 402 
2.  1989 
3.  3250 
4.  825 
5.  4000

Answer 1

(i) The square root of \(402\) can be calculated by long division method as follows. 

	20
2	\(\bar4\bar0\bar2\) \(-4\)
4	02 00
	2

The remainder is \(2\). 
It represents that the square of \(20\) is less than \(402\) by \(2\).
Therefore, a perfect square will be obtained by subtracting \(2\) from the given number \(402\).
Therefore, required perfect square = \(402 - 2 = 400\)
And, \(\sqrt{400} = 20\)

---

(ii) The square root of \(1989\) can be calculated by long division method as follows.

	44
4	\(\bar1\bar9\bar8\bar9\) \(-16\)
84	389 336
	53

The remainder is \(53\). 
It represents that the square of \(44\) is less than \(1989\) by \(53\). 
Therefore, a perfect square will be obtained by subtracting \(53\) from the given number \(1989\).
Therefore, required perfect square = \(1989 - 53 = 1936\) 
And, \(\sqrt{1936} = 44\)

---

(iii) The square root of \(3250\) can be calculated by long division method as follows. 

	57
5	\(\bar3\bar2\bar5\bar0\) \(-25\)
107	750 749
	1

The remainder is \(1\).
It represents that the square of \(57\) is less than \(3250\) by \(1\). 
Therefore, a perfect square can be obtained by subtracting \(1\) from the given number \(3250\).
Therefore, required perfect square = \(3250 - 1 = 3249\) 
And, \(\sqrt{3249} = 57\)

---

(iv) The square root of \( 825\) can be calculated by long division method as follows. 

	28
2	\(\bar8\bar2\bar5\) -4
48	425 384
	41

The remainder is \(41\). 
It represents that the square of \(28\) is less than \(825\) by \(41\).
Therefore, a perfect square can be calculated by subtracting.

---

(v) \(4000\) We know that, if we subtract the remainder from the number, we get a perfect square. 

	63
6	\(\bar4\bar0\bar0\bar0\) \(-36\)
123	400 -369
	31

Here, we get remainder \(31\). 
Therefore \(31\) must be subtracted from \(4000\) to get a perfect square. 
\(\therefore\) \(4000 – 31 = 3969 \)