Question

Find the smallest number which when multiplied with 3600 will make the product a perfect cube. Further, find the cube root of the product.

Answer 1

On factorising 3600 into prime factors, we get:

\[3600 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5\]

On grouping the factors in triples of equal factors, we get:

\[3600 = \left\{ 2 \times 2 \times 2 \right\} \times 2 \times 3 \times 3 \times 5 \times 5\]

It is evident that the prime factors of 3600 cannot be grouped into triples of equal factors such that no factor is left over.
Therefore, 3600 is not a perfect cube.
However, if the number is multiplied by ( \[2 \times 2 \times 3 \times 5 = 60\]) , the factors can be grouped into triples of equal factors such that no factor is left over.
Hence, the number 3600 should be multiplied by 60 to make it a perfect cube.
Also, the product is given as: 

\[3600 \times 60 = \left\{ 2 \times 2 \times 2 \right\} \times 2 \times 3 \times 3 \times 5 \times 5 \times 60\]
\[ \Rightarrow 216000 = \left\{ 2 \times 2 \times 2 \right\} \times 2 \times 3 \times 3 \times 5 \times 5 \times \left( 2 \times 2 \times 3 \times 5 \right)\]
\[ \Rightarrow 216000 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 3 \times 3 \times 3 \right\} \times \left\{ 5 \times 5 \times 5 \right\}\

To get the cube root of the produce 216000, take one factor from each triple.
Cube root = \[2 \times 2 \times 3 \times 5 = 60\]

Hence, the required numbers are 60 and 60.