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# What is the smallest number by which 8192 must be divided so that quotient is a perfect cube ? Also, find the cube root of the quotient so obtained.

Given:

210125

To do:

We have to find the smallest number by which 8192 must be divided so that quotient is a perfect cube and find the cube root of the product.

Solution:

Prime factorisation of 8192 is,

$8192=2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2$

$=2^3\times2^3\times2^3\times2^3\times2$

Grouping the factors in triplets of equal factors, we see that $2$ is left.

In order to make 8192 a perfect cube, we have to divide it by $2$.

$8192\div2=2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2\div2$

$=2^3\times2^3\times2^3\times2^3\times2\div2$

$\sqrt[3]{4096}=\sqrt[3]{2^3\times2^3\times2^3\times2^3}$

$=2\times2\times2\times2$

$=16$

The smallest number by which 8192 must be divided so that the quotient is a perfect cube is 2 and the cube root of the quotient is 16.

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