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


(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, 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


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