Carcass wrote:
For a certain positive integer N, N^3 has exactly 13 unique factors. How many unique factors does N have?
A. 1
B. 2
C. 3
D. 4
E. 5
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