IrMO 2014 Problem 6

Each of the four positive integers $N,N +1,N +2,N +3$ has exactly six positive divisors. There are exactly $20$ dierent positive numbers which are exact divisors of at least one of the numbers. One of these is $27$ . Find all possible values of $N$ .(Both $1$ and $m$ are counted as divisors of the number $m$ .)