Primes number $n,n+2,n+6,n+8,n+12,n+14$ 
Find all natural number $n$ such that all the following numbers are primes : 
  $$n,\;\; n+2,\;\;n+6,\;\;n+8,\;\;n+12,\;\;n+14$$
  are all prime numbers


As $n$ is a prime, $n\ge 2$.
When $n=2$, $n+2=4$ is not prime.
When $n=3$, $n+6=9$ is not prime.
When $n=5$, all of them are prime.
$(n+6)-n\equiv 1$ (mod $5$)
$(n+2)-n\equiv 2$ (mod $5$)
$(n+8)-n\equiv 3$ (mod $5$)
$(n+14)-n\equiv 4$ (mod $5$)
At least one of these integers is a multiple of $5$. 
When $n>5$, at least one of the numbers is not prime.
The only possible answer is $5$.
