By what smallest number should we multiply 53240, so that the product becomes a perfect cube. Find the cube root of the number formed.


Given :

The given number is 53240.

To do :

We have to find the smallest number that should multiply 53240 to make it a perfect cube.

Solution :

To find the smallest number by which 53240 be multiplied to make it a perfect cube, we have to find the prime factors of it.

Prime factorisation of 53240 is,

$53240 = 2 \times 2 \times 2 \times 5 \times 11 \times 11 \times 11$

$= (2 \times 2 \times 2) \times 5 \times (11 \times 11 \times 11)$

$= 2^3 \times 5 \times 11^3$.

As we can see, the given number is a product of 2 cube, 11 cube and 5. If we multiply the given number by 5 square it becomes a product of 2 cube, 5 cube and 11 cube. 

$53240 \times 5^2=  2^3 \times 5 \times 11^3\times 5^2$.

$53240 \times 25 = 2^3 \times 5^3 \times 11^3$.

$1331000 = (2\times 5\times 11)^3$.

$1331000 = 110^3$.

This implies,

Cube root of 1331000 is 110.

Therefore, the smallest number that has to be multiplied to make 53240 a perfect cube is 25 and the cube root of 1331000 is 110.

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Updated on: 10-Oct-2022

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