Carcass wrote:
If N is a positive integer and N^2 has 15 positive factors, how many positive factors can N have?
A. 5 or 7 factors
B. 6 or 8 factors
C. 4 or 6 factors
D. 9 or 7 factors
E. 9 or 8 factors
IMPORTANT RULE:
If N = (p^a)(q^b)(r^c)..., where p, q, r,...(etc.) are prime numbers, then the total number of positive divisors of N is equal to (a+1)(b+1)(c+1)...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-------------------
Scanning the answer choices, I can see that there must be TWO ways in which N² can have 15 positive factors.
Since 15 = (3)(5), we can see that, if N² = (some prime^
2)(some other prime^
4), then ...
...the total number of divisors of N² = (
2+1)(
4+1) = (3)(5) = 15
If N² = (some prime^
2)(some other prime^
4), then N = (some prime^
1)(some other prime^
2)
If N = (some prime^
1)(some other prime^
2), then the number of positive divisors of N = (
1+1)(
2+1) = (2)(3) =
6ELIMINATE A, D and E
Also recognize that if N² = (some prime^
14), then ...
...the total number of divisors of N² = (
14+1) = 15
If N² = (some prime^
14), then N = (some prime^
7)
If N = (some prime^
7), then the number of positive divisors of N = (
7+1) =
8Answer: B