Question

By which smallest number must the following number be divided so that the quotient is a perfect cube?

1600

Answer 1

On factorising 1600 into prime factors, we get:

\[1600 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5\]

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

\[1600 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\} \times 5 \times 5\]

It is evident that the prime factors of 1600 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 1600 is a not perfect cube. However, if the number is divided by (\[5 \times 5 = 25\]), the factors can be grouped into triples of equal factors such that no factor is left over.

Thus, 1600 should be divided by 25 to make it a perfect cube.