* Find all positive integers $n$ such that the numbers $n+1, n+3, n+7, n+9, n+13, n+15$ are all prime.
The only positive integer that satisfies the condition is $n=4$. In fact, when $n=1$, the number $n+3=4$ is composite; when $n=2$, the number $n+7=9$ is composite; when $n=3$, the number $n+1=4$ is composite; when $n>4$, the numbers $n+1, n+3, n+7, n+9, n+15$ are all greater than 5, and among these 5 numbers, at least one is divisible by 5 (since the numbers $1,3,7,9,15$ leave remainders of $1,3,2,4,0$ when divided by 5, so the numbers $n+1, n+3, n+7, n+9, n+15$ modulo 5 will cover all remainders $0,1,2,3,4$). Thus, this number is composite. On the other hand, when $n=4$, we get the primes: $5,7,11,13,17,19$.