By what smallest number should we multiply 26244 so that the number becomes a perfect cube? Find the cube root of the number formed.

Given :

The given number is 26244.

To do :

We have to find the smallest number by which 26244 be multiplied to make it a perfect cube.

Solution :

To find the smallest number by which 26244 be multiplied to make it a perfect cube, we have to find the prime factors of it.

Prime factorisation of 26244 is,

$26244 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$

          $= (2 \times 2) \times  (3 \times 3 \times 3) \times (3 \times 3 \times 3) \times (3 \times 3)$

          $ =  2^2 \times 3^3 \times 3^3 \times 3^2 $

As we can see, the given number is a product of 2 square, 3 cubes, 3 cubes, and 3 square. If we multiply the given number by a product of 2 and 3 then it becomes a product of 2 cubes, 3 cubes, 3 cubes, and 3 cubes. 

$26244 \times 2 \times 3 = 2^2 \times 3^3 \times 3^3 \times 3^2 \times 2 \times 3$

$26244 \times 6 = 2^3 \times 3^3 \times 3^3 \times 3^3$

$157464 = (2\times 3 \times 3\times 3)^3$.

$157464 = 54^3$.

This implies,

The cube root of 157464 is 54.

Therefore, the smallest number that has to be multiplied to make 26244 a perfect cube is 6 and the cube root of 154764 is 54.