A positive integer n is of the form n = 2 α 3 β , where α ≥ 1 , β ≥ 1 . If n has 12 positive divisors and 2n has 15 positive divisors, then the number of positive divisors of 3n is

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n = 2 α . 3 β ​ no. of divisors = ( α + 1 ) ( β + 1 ) = 12 ​ 2 n = 2 α + 1 3 β ​ ​ ⇒ α + 1 α + 2 ​ = 4 5 ​ ⇒ 4 α + 8 = 5 α + 5 ⇒ α = 3 ​ ⇒ β = 2 ⇒ 3 n = 2 3 3 3 ​ No. of divisors = ( 3 + 1 ) ( 3 + 1 ) = 16

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