If n has 15 positive divisors, inclusive of 1 and n, then wh

I didn't notice the case of 16 before great explanation. How would you rate this problem on close to 650 or above that?

Well, ~650 I guess.

I worked on the problem using the concept of finding number of positive factors of a given number.
If a given number N can be written in its prime factorized form as
N = \(a^{p}\)*\(b^{q}\)*\(c^{r}\).....,
then number of positive factors of a given number N = (p+1)(q+1)(r+1).....
so if number of factors of N = 15, then 15 can be written as
Case 1: (14+1). Thus here p = 14, i.e. N = \(3^{14}\). 3N = 3*\(3^{14}\) = \(3^{15}\) and will have 15+1 = 16 factors
Case 2: 3*5. Thus here p = 2 and q =4. N = \(a^{2}\)*\(b^{4}\). Thus 3N = \(3^{1}\)*\(a^{2}\)*\(b^{4}\), having number of factors as (1+1)(2+1)(4+1) = 30
or N = \(3^{2}\)*\(b^{4}\), thus 3N = \(3^{3}\)*\(b^{4}\), having number of factors as (3+1)(4+1) = 20
or N = \(a^{2}\)*\(3^{4}\), thus 3N = \(a^{2}\)*\(3^{5}\), having number of factors as (2+1)(5+1) = 18

Only 20 and 30 are satisfying thus answer is Option B

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