Here is the explanation to the answer provided.

But in the question it is asked to calculate the largest factor of `n`

which is actually `12`

. I don’t understand why we are considering the least factor `6`

as the answer. If you multiply `2^3`

with `6^3`

, the result (i.e `n^3`

=`12^3`

) is actually a multiple of `24`

.

n^3 is divisible by 24 means ==> n^3 / 24 = integer (assume integer as p)

n^3 / 24 ==> p

24 prime factoriz…= 2^3 * 3

n^3 ==> 24 * p

n^3 ==> 2^3 * 3 * p

taking cube roots on both sides

(n^3)^1/3 ==> ( 2^3 * 3 * p )^1/3

n ==> (2^3)^1/3 * 3^1/3 * p^1/3

to get an integer we need sub p with 3^2

n ==> 2 * 3^1/3 * (3^2)^1/3

n ==> 2 * 3^(1/3+2/3)

n ==> 2* 3^3/3

n==> 2 * 3^1

n ==> 2*3 == 6

To get an integer we can also substitute `p`

with `(2^3 * 3^2)`

```
n ==> 2 * 3^(1/3) * (2^3 * 3^2)^(1/3)
or, n ==> 2 * 3^(1/3) * 2 * 3^(2/3)
or, n ==> 2 * 2 * 3^(1/3+2/3)
or, n ==> 2 * 2 * 3 == 12 (Option E)
```

The question asked for **must be**