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) 675 (ii) 100
Answer» (i) 675 = 3 × 3 × 3 × 5 × 5 Here, two 5s are left which are not in a triplet. To make 675 a cube, one more 5 is required. Then, we obtain 675 × 5 = 3 × 3 × 3 × 5 × 5 × 5 = 3375 is a perfect cube. Hence, the smallest natural number by which 675 should be multiplied to make it a perfect cube is 5. (ii) 100 = 2 × 2 × 5 × 5 Here, two 2s and two 5s are left which are not in a triplet. To make 100 a cube, we require one more 2 and one more 5. Then, we obtain 100 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5 = 1000 is a perfect cube Hence, the smallest natural number by which 100 should be multiplied to make it a perfect cube is 2 × 5 = 10.

Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.(i) 675 (ii) 100

Answer»

(i) 675 = 3 × 3 × 3 × 5 × 5

Here, two 5s are left which are not in a triplet. To make 675 a cube, one more 5 is required.

Then, we obtain

675 × 5 = 3 × 3 × 3 × 5 × 5 × 5 = 3375 is a perfect cube.

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

(ii) 100 = 2 × 2 × 5 × 5

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

Then, we obtain

100 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5 = 1000 is a perfect cube

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