Q. 23.9( 117 Votes )

Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube

(i) 243 (ii) 256

(iii) 72 (iv) 675

(v) 100

Answer :

(i) 243 = 3 x 3 x 3 x 3 x 3

Here, we can find that two 3s are left which are not forming a triplet. To make 243 a cube, one more 3 is required.


Hence,


243 x 3 = 3 x 3 x 3 x 3 x 3 x 3 = 729


729 is a perfect cube.


Therefore, the smallest natural number by which 243 should be multiplied to make it a perfect cube is 3


(ii) 256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2


Here, we can see that two 2s are left which are not forming a triplet. To make 256 a cube, one more 2 is required


Hence,


256 x 2 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 512


Whereas 512 is a perfect cube.


Therefore, the smallest natural number by which 256 should be multiplied to make it a perfect cube is 2


(iii) 72 = 2 x 2 x 2 x 3 x 3


Here, we can see that two 3s are left which are not forming a triplet. In order to make 72 a cube, one more 3 is required


Hence,


72 x 3 = 2 x 2 x 2 x 3 x 3 x 3 = 216


We know that 216 is a perfect cube


Therefore, the smallest natural number by which 72 should be multiplied to make it a perfect cube is 3


(iv) 675 = 3 x 3 x 3 x 5 x 5


Here, we can see that two 5s are left which are not forming a triplet. To make 675 a cube, one more 5 is needed


Hence,


675 x 5 = 3 x 3 x 3 x 5 x 5 x 5 = 3375


We know that 3375 is a perfect cube.


Therefore, the smallest natural number by which 675 should be multiplied to make it a perfect cube is 5


(v) 100 = 2 x 2 x 5 x 5


Here, we can see that two 2s and two 5s are left which are not forming a triplet. To make 100 a cube, we require one more 2 and one more 5


Hence,


100 x 2 x 5 = 2 x 2 x 2 x 5 x 5 x 5 = 1000


We know that 1000 is a perfect cube.


Therefore, the smallest natural number by which 100 should be multiplied to make it a perfect cube is 2 x 5 = 10


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