Question

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

 2560

Answer 1

On factorising 2560 into prime factors, we get:

\[2560 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5\]

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

\[2560 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\} \times 5\]

It is evident that the prime factors of 2560 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 2560 is a not perfect cube. However, if the number is multiplied by  \[5 \times 5 = 25\] the factors can be grouped into triples of equal factors such that no factor is left over.

Thus, 2560 should be multiplied by 25 to make it a perfect cube.