Question

By what smallest number should $3600$ be multiplied so that the quotient is a perfect cube.

Also find the cube root of the quotient.

Answer 1

Step 1 : Finding prime factors by using prime factorization method

Let us first determine the factors by the method of prime factorization.

Factors of $3600=2\times 2\times 2\times 2\times 3\times 3\times 5\times 5$

On grouping and pairing the obtained factors we get $=\left(2\times 2\times 2\right)\times 2\times 3\times 3\times 5\times 5$

Factors $2$, $3$ and $5$ are unpaired prime factors.

$\therefore 3600$ is not a perfect cube.

Step 2 : Finding the smallest number

The smallest number it should be multiplied to get a perfect cube is $2\times 2\times 3\times 5=60$

Hence,

Product $=3600\times 60=216000$

Factors of $216000=2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 5\times 5\times 5$

$$\begin{gathered} \sqrt[3]{216000} \\ =\sqrt[3]{2^3\times 2^3\times 3^3\times 5^3} \\ =2\times 2\times 3\times 5 \\ =60 \end{gathered}$$

Hence, $3600$ must be multiplied by $60$, to make it a perfect cube.

And also the cube root of the quotient is $60$