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