Question

What is the smallest number by which the following number must be multiplied, so that the products is perfect cube?

 35721

Answer 1

On factorising 35721 into prime factors, we get:

\[35721 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 7 \times 7\]

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

 

\[35721 = \left\{ 3 \times 3 \times 3 \right\} \times \left\{ 3 \times 3 \times 3 \right\} \times 7 \times 7\]

It is evident that the prime factors of 35721 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 35721 is a not perfect cube. However, if the number is multiplied by 7, the factors be grouped into triples of equal factors such that no factor is left over.

Thus, 35721 should be multiplied by 7 to make it a perfect cube.