Question

Find the smallest number by which 53240 must be multiplyed to obtain a perfect cube.​

Answer 1

Answer:

If we divide the number by 5, then the prime factorisation of the quotient will not contain 5. Hence the smallest number by which 53240 should be divided to make it a perfect cube is 5. The perfect cube in that case is = 10648.

Answer 2

First, we need to prime factorization of 53240.

$\begin{array}{r | l} 2 & 53240 \\ \cline{2-2} 2 & 26620 \\ \cline{2-2} 2 &13310 \\ \cline{2-2} 5 & 6655 \\ \cline{2-2} 11 &1331 \\ \cline{2-2} 11 & 121 \\ \cline{2-2} & 11 \\ \end{array}$

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

Now let's group the prime factors in groups of 3.

$\sf 53240 =>( 2\times 2\times 2)\times 5\times (11 \times 11\times 11)$

The prime factor 5 does not appear in any of the three groups.

∴53240 is not a perfect cube.

According to the prime factorization, 5 is only once.

So we need to multiply 25 (5×5) to 53240 to obtain a perfect cube.

$\sf 53240\times 25 =1331000$

$\sf 1331000 => 2\times 2\times 2\times 5 \times 5 \times 5\times 11\times 11\times 11$

Let's group the numbers in the group of 3.

$\sf 1331000 => (2\times 2\times 2)\times (5 \times 5 \times 5)\times (11\times 11\times 11)$

∴1331000 is a perfect cube.

∴The smallest number by which 53240 must be multiplied to obtain a perfect cube is 25.