Explanation:

Any number of the form p^aq^br^c will have (a + 1) (b + 1) (c + 1) factors, where p, q, r are prime.
N² has 15 factors.
Now, 15 can be written as 1 * 15 or 3 * 5.
If we take the underlying prime factorization of N² to be paqb, then it should have (a + 1) (b + 1) factors.
So, N can be of the form
p¹⁴ or p²q⁴
p¹⁴ will have (14 + 1) = 15 factors
p²q⁴ 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.
Hence, N can have 6 or 8 factors.