For each positive integer n, define \varphi(n) to be the number of positive divisors of n. For example, \varphi(10)=4, since 10 has 4 positive divisors, namely \{1,2,5,10\}.
Suppose n is a positive integer such that \varphi(2 n)=6. Determine the minimum possible value of \varphi(6 n).