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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.