Any number of the form p^{a}q^{b}r^{c} will have (a + 1) (b + 1)(c + 1) factors, where p, q, r are prime. (This is a very important idea)
N^{2} has 15 factors.
Now, 15 can be written as 1 * 15 or 3 * 5.
If we take the underlying prime factorization of N^{2} to be p^{a}q^{b}, then it should have (a + 1) (b + 1) factors. So, N can be of the form
p^{14} or p^{2}q^{4}
p^{14} will have (14 + 1) = 15 factors
p^{2}q^{4} will have (2 + 1) * (4 + 1) = 15 factors.
Importantly, these are the only two possible prime factorizations that can result in a number having 15 factors.
Correct Answer: 6 or 8 factors.