Question

By which smallest number must the following number be divided so that the quotient is a 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 divided by 

\[5 \times 5 = 25\] , the factors can be grouped into triples of equal factors such that no factor is left over.

Thus, 675 should be divided by 25 to make it a perfect cube.