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A positive integer n is of the form n=2^...

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`

Text Solution

Verified by Experts

The correct Answer is:
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`
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