Question

Which of the following number is cube of negative integer - 2197.

Answer 1

In order to check if a negative number is a perfect cube, first check if the corresponding positive integer is a perfect cube. Also, for any positive integer m, -m³ is the cube of - m.

On factorising 2197 into prime factors, we get: 

\[2197 = 13 \times 13 \times 13\]

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

\[2197 = \left\{ 13 \times 13 \times 13 \right\}\]

It is evident that the prime factors of 2197 can be grouped into triples of equal factors and no factor is left over. Therefore, 2197 is a perfect cube. This implies that - 2197 is also a perfect cube.

 

Now, collect one factor from each triplet and multiply, we get 13.
This implies that 2197 is a cube of 13.
Thus, - 2197 is the cube of - 13.