Question

A positive integer `n` is of the form `n=2^(alpha)3^(beta)`, where `alpha ge 1`, `beta ge 1`. If `n` has `12` positive divisors and `2n` has `15` positive divisors, then the number of positive divisors of `3n ` is
A. `15`
B. `16`
C. `18`
D. `20`

Answer 1

Correct Answer - B
`(b)` `n=2^(alpha)*3^(beta)`
No. of divisors `=(alpha+1)(beta+1)=12`
`2n=2^(alpha+1)3^(beta)`
No.of divisors `=(alpha+2)(beta+1)=15`
`implies(alpha+2)/(alpha+1)=(5)/(4)`
`implies4alpha+8=5alpha+5impliesalpha=3`
`impliesbeta=2implies3n=2^(3)3^(3)`
No. of divisors `=(3+1)(3+1)=16`