IMPORTANT:
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40

-------now onto the question------------------------------------------------

In the above rule, notice that total number of divisors is a PRODUCT [e.g., (4+1)(3+1)(1+1) =(5)(4)(2) = 40]

In this question, we're told that there are 13 divisors.
There's only ONE WAY that the number 13 can be written as a product: 1 x 13

So, a number with 13 positive divisors must have a prime factorization that looks like this: prime^12

So, if N³ is equivalent to prime^12, then N must equal prime^4, since (prime^4)^3 = prime^12

For example, if N = 2^4, then N = 16
Notice that, if N = 2^4, then N³ = (2^4)^3 = 2^12
And, from the above rule, if N³ = 2^12, then the number of positive divisors of N³ = (12+1) = 13

How many unique factors does N have?
If N = 16 (or any other prime^4), then the factors of 16 = {1, 2, 4, 8, 16}
There are 5 factors.

Answer: E