Question

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

107811

Answer 1

On factorising 107811 into prime factors, we get:

\[107811 = 3 \times 3 \times 3 \times 3 \times 11 \times 11 \times 11\]

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

\[107811 = \left\{ 3 \times 3 \times 3 \right\} \times 3 \times \left\{ 11 \times 11 \times 11 \right\}\]

It is evident that the prime factors of 107811 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 107811 is a not perfect cube. However, if the number is multiplied by  \[3 \times 3 = 9\], the factors be grouped into triples of equal factors such that no factor is left over.

Thus, 107811 should be multiplied by 9 to make it a perfect cube.