Question

7. By what number should each of the following numbers be 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​

Answer 1

Explanation:

(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 

(ii) 3675  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 

(iii) 605 

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 

(iv) 2880 

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 

(v) 4056 

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  (vi) 3468 

3468 = (2 × 2) × 3 × (17 × 17)

  In the above factors only 3 are unpaired  So, multiply 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.