Question

For of the non-perfect cubes in Q. No. 20 find the smallest number by which it must be divided so that the quotient is a perfect cube.

Answer 1

The only non-perfect cube in question number 20 is 243.

On factorising 243 into prime factors, we get: \[243 = 3 \times 3 \times 3 \times 3 \times 3\] On grouping the factors in triples of equal factors, we get:

\[243 = \left\{ 3 \times 3 \times 3 \right\} \times 3 \times 3\]It is evident that the prime factors of 243 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 243 is not a perfect cube. However, if the number is divided by ( \[3 \times 3 = 9\]), the factors can be grouped into triples of equal factors such that no factor is left over.

Thus, 243 should be divided by 9 to make it a perfect cube.