We are tasked with finding all positive integers $n$ such that $n$, $n+2$, and $n+4$ are all prime numbers. These three numbers form what is known as a "prime triplet."

Step 1: Analyze properties of prime numbers in this form

Let $n$, $n+2$, and $n+4$ be prime numbers.

• Since $n$, $n+2$, and $n+4$ are consecutive odd numbers (because any even $n$ greater than 2 would make $n+2$ or $n+4$ even and thus non-prime), we focus on $n$ being an odd number.
• If $n$ is odd, then one of the numbers in $n$, $n+2$, and $n+4$ must be divisible by 3. This is because every third integer is divisible by 3.

Step 2: Check divisibility by 3

We now consider the three cases for divisibility by 3:

• Case 1: $n$ is divisible by 3. Then $n = 3$ (since 3 is the only prime divisible by 3). In this case, we check $n+2 = 5$ and $n+4 = 7$, which are both prime. Thus, $n = 3$ works.

• Case 2: $n+2$ is divisible by 3. If $n+2$ is divisible by 3 and greater than 3, it cannot be prime. Therefore, there are no solutions in this case.

• Case 3: $n+4$ is divisible by 3. Similarly, if $n+4$ is divisible by 3 and greater than 3, it cannot be prime. Hence, there are no solutions in this case either.

Step 3: Conclusion

The only solution for which all three numbers $n$, $n+2$, and $n+4$ are prime is when $n = 3$.

Thus, the only positive integer $n$ such that $n$, $n+2$, and $n+4$ are all prime is: $\boxed{3}$

Would you like more details or have any questions? Here are five related questions to expand on this topic:

1. Can there be any prime triplets beyond $n, n+2, n+4$?
2. Why is 3 the only prime divisible by 3?
3. Are there other consecutive prime numbers similar to twin primes or prime triplets?
4. What are twin primes, and do they follow similar divisibility patterns?
5. How can the prime number theorem help in understanding patterns of prime numbers?

Tip: Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves; studying their properties helps in understanding number theory.