Question

By which smallest number must the following number be divided so that the quotient is a 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 group 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 divided by 3, the factors can be grouped into triples of equal factors such that no factor is left over.

Thus, 107811 should be divided by 3 to make it a perfect cube.