Find the smallest number by which each of the following numbers must b

1.	Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.(i) 243 (ii) 256 (ii) 72
Answer» (i) 243 = 3 × 3 × 3 × 3 × 3 Here, two 3s are left which are not in a triplet. To make 243 a cube, one more 3 is required. In that case, 243 × 3 = 3 × 3 × 3 × 3 × 3 × 3 = 729 is a perfect cube. Hence, the smallest natural number by which 243 should be multiplied to make it a perfect cube is 3. (ii) 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 Here, two 2s are left which are not in a triplet. To make 256 a cube, one more 2 is required. Then, we obtain 256 × 2 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512 is a perfect cube. Hence, the smallest natural number by which 256 should be multiplied to make it a perfect cube is 2. (iii) 72 = 2 × 2 × 2 × 3 × 3 Here, two 3s are left which are not in a triplet. To make 72 a cube, one more 3 is required. Then, we obtain 72 × 3 = 2 × 2 × 2 × 3 × 3 × 3 = 216 is a perfect cube. Hence, the smallest natural number by which 72 should be multiplied to make it a perfect cube is 3.

Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.(i) 243 (ii) 256 (ii) 72

Answer»

(i) 243 = 3 × 3 × 3 × 3 × 3

Here, two 3s are left which are not in a triplet. To make 243 a cube, one more 3 is required.

In that case, 243 × 3 = 3 × 3 × 3 × 3 × 3 × 3 = 729 is a perfect cube.

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

(ii) 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Here, two 2s are left which are not in a triplet. To make 256 a cube, one more 2 is required.

Then, we obtain

256 × 2 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512 is a perfect cube.

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

(iii) 72 = 2 × 2 × 2 × 3 × 3

Here, two 3s are left which are not in a triplet. To make 72 a cube, one more 3 is required.

Then, we obtain

72 × 3 = 2 × 2 × 2 × 3 × 3 × 3 = 216 is a perfect cube.

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