8. Question 2.
Find the least number which must be subtracted from the following numbers
to make them a perfect square :
9. (i) 2361
(ii) 194491
(iii) 26535
(iv) 16160
(v) 4401624
Solution:
(i) 2361
Finding the square root of 2361
We get 48 as quotient and remainder = 57
∴ To make it a perfect square, we have to subtract 57 from 2361
∴ Least number to be subtracted = 57
(ii) 194491
Finding the square root of 194491
We get 441 as quotient and remainder = 10
∴ To make it a perfect square, we have to subtract 10 from 194491
∴ Least number to be subtracted = 10
(iii) 26535
Finding the square root of 26535
We get 162 as quotient and 291 as remainder
∴ To make it a perfect square, we have to subtract 291 from 26535
∴ Least number to be subtracted = 291
(iv)16160
Finding the square root of 16160
10. We get 127 as quotient and 31 as remainder
∴ To make it a perfect square, we have to subtract 31 from 16160
∴ Least number to be subtracted = 31
(v) 4401624
Find the square root of 4401624
We get 2098 as quotient and 20 as remainder
∴ To make it a perfect square, we have to subtract 20 from 4401624
∴ Least number to be subtracted = 20
Question 3.
Find the least number which must be added to the following numbers to make them
a perfect square :
(i) 5607
(ii) 4931
(iii) 4515600
(iv) 37460
(v) 506900
Solution:
(i) 5607
Finding the square root of 5607, we see that 742
= 5607- 131 =5476 and 752
= 5625
∴ 5476 < 5607 < 5625
∴ 5625 – 5607 = 18 is to be added to get a perfect square
∴ Least number to be added = 18
11. (ii) 4931
Finding the square root of 4931, we see that 702
= 4900
∴ 712
= 5041 4900 <4931 <5041
∴ 5041 – 4931 = 110 is to be added to get a perfect square.
∴ Least number to be added =110
(iii) 4515600
Finding the square root of 4515600, we see
that 21242
= 4511376
and 2 1 252
= 45 1 56 25
∴ 4511376 <4515600 <4515625
∴ 4515625 – 4515600 = 25 is to be added to get a perfect square.
∴ Least number to be added = 25
(iv) 37460
Finding the square root of 37460
that 1932
= 37249, 1942
= =37636
∴ 37249 < 37460 < 37636
∴ 37636 – 37460 = 176 is to be added to get a perfect square.
∴ Least number to be added =176
(v) 506900
12. Finding the square root of 506900, we see that
7112
= 505521, 7122
= 506944
∴ 505521 < 506900 < 506944
∴ 506944 – 506900 = 44 is to be added to get a perfect square.
∴ Least number to be added = 44
Question 4.
Find the greatest number of 5 digits which is a perfect square.
Solution:
Greatest number of 5-digits = 99999 Finding square root, we see that 143 is left as
remainder
∴ Perfect square = 99999 – 143 = 99856 If we add 1 to 99999, it will because a
number of 6 digits
∴ Greatest square 5-digits perfect square = 99856
Question 5.
Find the least number of four digits which is a perfect square.
Solution:
Least number of 4-digits = 10000
Finding square root of 1000
We see that if we subtract 39
From 1000, we get three digit number
∴ We shall add 124 – 100 = 24 to 1000 to get a
13. perfect square of 4-digit number
∴ 1000 + 24 = 1024
∴ Least number of 4-digits which is a perfect square = 1024
Question 6.
Find the least number of six-digits which is a perfect square.
Solution:
Least number of 6-digits = 100000
Finding the square root of 100000, we see that if we subtract 544, we get a perfect
square of 5-digits.
So we shall add
4389 – 3900 = 489
to 100000 to get a perfect square
Past perfect square of six digits= 100000 + 489 =100489
Question 7.
Find the greatest number of 4-digits which is a perfect square.
Solution:
Greatest number of 4-digits = 9999
Finding the square root, we see that 198 has been left as remainder
∴ 4-digit greatest perfect square = 9999 – 198 = 9801
Question 8.
A General arranges his soldiers in rows to form a perfect square. He finds that in
doing so, 60 soldiers are left out. If the total number of soldiers be 8160, find the
number of soldiers in each row.
Solution:
Total number of soldiers = 8160 Soldiers left after arranging them in a square = 60
14. ∴ Number of soldiers which are standing in a square = 8160 – 60 = 8100
Question 9.
The area of a square field is 60025 m2
. A man cycle along its boundry at 18 km/hr. In
how much time will be return at the starting point.
Solution:
Area of a square field = 60025 m2
Question 10.
The cost of levelling and turfing a square lawn at Rs. 250 per m2
is Rs. 13322.50.
Find the cost of fencing it at Rs. 5 per metre ?
Solution:
15. Cost of levelling a square field = Rs. 13322.50
Rate of levelling = Rs. 2.50 per m2
and perimeter = 4a = 4 x 73 = 292 m Rate of fencing the field = Rs. 5 per m
∴ Total cost of fencing = Rs. 5 x 292 = Rs. 1460
Question 11.
Find the greatest number of three digits which is a perfect square.
Solution:
3-digits greatest number = 999
Finding the square root, we see that 38 has been left
∴ Perfect square = 999 – 38 = 961
∴ Greatest 3-digit perfect square = 961
Question 12.
Find the smallest number which must be added to 2300 so that it becomes a perfect
square.
Solution:
Finding the square root of 2300
We see that we have to add 704 – 700 = 4 to 2300 in order to get a perfect square
∴ Smallest number to be added = 4