question_answer 25)

                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. (i) 402                                    (ii) 1989                (iii) 3250                               (iv) 825 (v) 4000

Answer:

                (i) 402 We have  

	2            0
2	\[\overline{4}\]         \[\overline{02}\] ?4
40	02               ?00
	0

This shows that 202 is less than 402 by 2. This means, if we subtract the remainder from the number, we get a perfect square. So, the required least number is 2. Therefore, the required perfect square is \[402-2=400\] Hence, \[\sqrt{400}=20\]. (ii) 1989 We have  

	4            4
4	\[\overline{19}\]        \[\overline{89}\] ?16
84	3             89 ?3            36
	53

This shows that \[{{44}^{2}}\] is less than 1989 by 53. This means that if we subtract the remainder from the number, we get a perfect square. So, the required least number is 53. Therefore, the required perfect square is 1989 - 53 = 1936. Hence, \[\sqrt{1936}=44\]. (iii) 3250 We have

	5            7
5	\[\overline{32}\]        \[\overline{50}\] ?25
107	7             50 ?7            49
	1

This shows that \[{{57}^{2}}\] is less than 3250 by 1. This means if we subtract the remainder from the number, we get a perfect square. So, the required least number is 1. Therefore, the required perfect square is \[3250-1=3249\] Hence, \[\sqrt{3249}\,=57\]. (iv) 825 We have

	2            8
2	\[\overline{8}\]         \[\overline{25}\] ?4
48	4            25 ?3           84
	41

  This shows that 282 is less than 825 by 41. This means if we subtract the remainder from the number, we get a perfect square. So, the required least number is 41. Therefore, the required perfect square is \[825-41=784\] Hence, \[\sqrt{784}=28\]. (v) 4000 We have  

	63
2	\[\overline{40}\]         \[\overline{00}\] ?36
123	4            00 ?3          69
	31

This shows that 632 is less than 4000 by 31.                                  This means if we subtract the remainder from the number, we get a perfect square. So, the required least number is 31. Therefore, the required perfect square is \[4000-31=3969\]. Hence, \[\sqrt{3969}=63\].