Question

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

1323

Answer 1

On factorising 1323 into prime factors, we get:

\[1323 = 3 \times 3 \times 3 \times 7 \times 7\]

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 1323 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 1323 is a not perfect cube. However, if the number is multiplied by 7, the factors can be grouped into triples of equal factors and no factor will be left over.

Thus, 1323 should be multiplied by 7 to make it a perfect cube.