Suppose n is a positive integer such that 6n has exactly 9 positive divisors. How many prime numbers are divisors of 6n?
6=2131let n=∞∏k=0pnkkwhere pk is the kth prime6n=2n2+1⋅3n3+1⋅∞∏k=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