If n has 15 positive divisors, inclusive of 1 and n, then which of the following could be the number of divisors of 3n?
I. 20
II. 30
III. 40
Answers:
A) II only
B) I and II only
C) I and III only
D) II and III only
E) I, II and III only
n has 15 positive divisors --&gt; n=p^14 (the # of factors   (14+1)= 
<gadget id="calculator">  (14+1)</gadget>
<output>15</output>
15) or n=p^2q^4 (the # of factors (2+1)(4+1)=15).   If neither p nor q is 3, then:   3n=3p^14 will have (1+1)(14+1)=30 factors.   3n=3p^2q^4 will have (1+1)(2+1)(4+1)=30 factors.   If p = 3, then:   3n=3p^14 =3^15 will have   (15+1)= 
<gadget id="calculator">  (15+1)</gadget>
<output>16</output>
16 factors.   3n=3p^2q^4 = 3^3 p^4 will have (3+1)(4+1)=20 factors.   If q = 3, then:   3n=3p^2 =q^4 will have(2+1) (5+1)=18 factors.   Answer: I and II only.
<result>B</result>
If n has 15 positive divisors, inclusive of 1 and n, then which of the following could be the number of divisors of 3n?
I. 20
II. 30
III. 40