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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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