Find all positive integers $n$ such that $n$, $n+2$, and $n+4$ are all prime.

We are tasked with finding all positive integers $ n $ such that $ n $, $ n+2 $, and $ n+4 $ are all prime numbers. Let's analyze this step by step.

### Step 1: Explore small values of $ n $
First, let's check small values of $ n $ to see if we can find any primes that fit the condition.

- For $ n = 3 $:
  - $ n = 3 $, $ n+2 = 5 $, and $ n+4 = 7 $. All three numbers are prime.
  - Therefore, $ n = 3 $ is a solution.

### Step 2: General Analysis
Now let's see if there are any other solutions.

We are looking for three numbers $ n $, $ n+2 $, and $ n+4 $ that are all prime. Consider the following points:

- One of the numbers must be divisible by 3, since among any three consecutive numbers $ n $, $ n+2 $, and $ n+4 $, one of them must be divisible by 3. This follows from the fact that every third number is divisible by 3.
  
  - If $ n $ is divisible by 3, then $ n = 3 $ (since any higher multiple of 3 is not prime).
  - If $ n+2 $ is divisible by 3, then $ n+2 = 3 $, which implies $ n = 1 $. However, $ n = 1 $ is not prime, so this case is not possible.
  - If $ n+4 $ is divisible by 3, then $ n+4 = 3 $, which implies $ n = -1 $, but this is not a positive integer.

Thus, the only possibility for $ n $ being divisible by 3 is when $ n = 3 $.

### Step 3: Conclusion
The only solution is $ 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 further details or clarifications?

Here are five related questions to explore further:
1. What are the first 10 prime numbers?
2. Can there be any sequence of 4 consecutive odd primes?
3. What is the largest known prime number, and how is it discovered?
4. Why is 2 the only even prime number?
5. How can we prove that there are infinitely many prime numbers?

**Tip:** Prime numbers greater than 3 can be expressed in the form $ 6k \pm 1 $.

Discover how to find all positive integers n such that n, n+2, and n+4 are prime numbers. This detailed solution explores number theory concepts and uses the divisibility rule for 3 to conclude that n = 3 is the only solution. Ideal for students in Grades 9-12.

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