Question

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

675

Answer 1

On factorising 675 into prime factors, we get:

\[675 = 3 \times 3 \times 3 \times 5 \times 5\]

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

 

\[675 = \left\{ 3 \times 3 \times 3 \right\} \times 5 \times 5\]

It is evident that the prime factors of 675 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 675 is a not perfect cube. However, if the number is multiplied by 5, the factors can be grouped into triples of equal factors and no factor will be left over.

Thus, 675 should be multiplied by 5 to make it a perfect cube.