Question

List all the perfect cubes between 1000 and 3000 that are odd

Answer 1

1000 is exactly 10^3

Since
`\sqrt[3]{3000} \approx 14.42`, the cube nearest to 3000 (without exceeding it) is 14^3.

Now, the cube of an even number is still even, the cube of an odd number is still odd. In fact, an even number contains at least one factor 2 in its prime factorization, and when you cube it, the 2 will still be there.

On the other hand, an odd number has no factor 2 in its factorization, and since cubing a number means multiplying it by itself, that factor 2 will not appear.

So, we're looking for cubes of numbers between 10 and 14, and we want the cubes to be odd (which implies the roots to be odd).

This means that the only feasible choices are 11^3 and 13^3, which are respectively

`11^3 = 1331,\quad 13^3 = 2197`

Just to prove that these are the only two possibilities: 9 is not enough, because
`9^3 = 729`, and 15 is too much, because
`15^3 = 3375`.

Answer 2

There are only two perfect cubes between 1000 and 3000 which are odd.

1331 (11^3)

2197 (13^3)

Hope it helps :)