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

(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

5