Question

By which smallest number must the following number be divided so that the quotient is a 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 divided by (\[7 \times 7 = 49\]), the factors can be grouped into triples of equal factors such that no factor is left over.

Thus, 35721 should be divided by 49 to make it a perfect cube.