By what number should each of the following numbers by multiplied to get a perfect square in each case? Also find the number whose square is the new number.
(i) 8820 (ii) 3675
(iii) 605 (iv) 2880
(v) 4056 (vi) 3468
(vii) 7776
(i) 8820
8820 = (2 × 2) × (3 × 3) × (7 × 7) × 5
In the above factors only 5 is unpaired
So, multiply the number with 5 to make it paired
Again,
8820 × 5 = 2 × 2 × 3 × 3 × 7 × 7 × 5 × 5
= (2 × 2) × (3 × 3) × (7 × 7) (5 × 5)
= (2 × 3 × 7 × 5) × (2 × 3 × 7 × 5)
= 210 × 210
= (210)2
So, the product is the square of 210
3675 = (5 × 5) × (7 × 7) × 3
In the above factors only 3 is unpaired
So, multiply the number with 3 to make it paired
Again,
3675 × 3 = 5 × 5 × 7 × 7 × 3 × 3
= (5 × 5) × (7 × 7) × (3 × 3)
= (3 × 5 × 7) × (3 × 5 × 7)
= 105 × 105
= (105)2
So, the product is the square of 105
605 = 5 × (11 × 11)
In the above factors only 5 is unpaired
So, multiply the number with 5 to make it paired
Again,
605 × 5 = 5 × 5 × 11 × 11
= (5 × 5) × (11 × 11)
= (5 × 11) × (5 × 11)
= 55 × 55
= (55)2
So, the product is the square of 55
2880 = 5 × (3 × 3) × (2 × 2) × (2 × 2) × (2 × 2)
In the above factors only 5 is unpaired
So, multiply the number with 5 to make it paired
Again,
2880 × 5 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5
= (2 × 2) × (2 × 2) × (2 × 2) (3 × 3) × (5 × 5)
= (2 × 2 × 2 × 3 × 5) × (2 × 2 × 2 × 3 × 5)
= 120 × 120
= (120)2
So, the product is the square of 120
4056 = (2 × 2) × (13 × 13) × 2 × 3
In the above factors only 2 and 3 are unpaired
So, multiply the number with 6 to make it paired
Again,
4056 × 6 = 2 × 2 × 13 × 13 × 2 × 2 × 3 × 3
= (2 × 2) × (13 × 13) × (2 × 2) (3 × 3)
= (2 × 2 × 3 × 13) × (2 × 2 × 3 × 13)
= 156 × 156
= (156)2
So, the product is the square of 156
3468 = (2 × 2) × 3 × (17 × 17)
In the above factors only 3 are unpaired
So, mulityply the number with 3 to make it paired
3468 × 3 = (2 × 2) × (3 × 3) × (17 × 17)
= (2 × 3 × 17) × (2 × 3 × 17)
= 102 × 102
= (102)2
So, the product is the square of 102
(vii) 7776
7776 = (2 × 2) × (2 × 2) × (3 × 3) × (3 × 3) × 2 × 3
In the above factors only 2 and 3 are unpaired
So, multiply the number with 6 to make it paired
Again,
7776 × 6 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
= (2 × 2) × (2 × 2) × (2 × 2) (3 × 3) × (3 × 3) × (3 × 3)
= (2 × 2 × 2 × 3 × 3 × 3) × (2 × 2 × 2 × 3 × 3 × 3)
= 216 × 216
= (216)2
So, the product is the square of 216