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Suppose n is a positive integer such that 6n has exactly 9 positive divisors.

How many prime numbers are divisors of 6n?

 Mar 27, 2019
 #1
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6=2131let n=k=0pnkkwhere pk is the kth prime6n=2n2+13n3+1k=3pnkk

 

6n thus has (n2+2)(n3+2)k=0(nk+1)=9 divisors

This is easily enough accomplished if n2=1, n3=1, nk=0,k>3, i.e. 6n=36=2232There are a total of 2 prime divisors

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 Mar 27, 2019

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