WHAT NUMBER SHOULD BE MULTIPLIED OR DIVIDED TO MAKE PERFECT CUBE

Problem 1 :

What least number is to be multiplied with 3087 in order to get a perfect cube?

Solution :

Prime factorization of 3087 is,

3087 = 3 × 3 × 7 × 7 × 7

= 7³ × 3²

Hence, the smallest number by which 3087 must be multiplied to obtain a perfect cube is 3.

Problem 2 :

What is the least number by which 12800 should be divided in order to get a perfect cube?

Solution :

12800 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5

= 2⁹ × 5²

12800 is divided by 25 perfect cube is 512.

So, the cube root is 8.

Problem 3 :

Find the smallest common multiple of 48, 72 and 32 that is a perfect cube. 

Solution :

LCM (48, 72, 32) = 2 x 2 x 2 x 3 x 2 x 3 x 2

= 288

Decomposing 288, we get

= 2 x 2 x 2 x 3 x 3 x 2 x 2

To make it as perfect cube, we need one more 2 and one more 3.

So, the smallest number to be multiplied is 6.

Problem 4 :

By which least number should 72000 be multiplied such that the result is a perfect cube?

Solution :

Decomposing 72000, we get 

72000 = 72 x 10 x 10 x 10

= 2 x 2 x 2 x 3 x 3 x 10 x 10 x 10

To group the values, we need one more 3.

So, 3 is the least number to be multiplied to make it is perfect cube.

Problem 5 :

By which least number should 171500 be divided such that the result is a perfect cube ?

Solution :

171500 = 5 x 7 x 7 x 7 x 5 x 2 x 5 x 2

= 5³ x 7³ x 2²

So, the smallest number to be multiplied to make it as prefect cube is 2.