Find all positive integers $ n$ such that $ 8^n \plus{} n$ is divisible by $ 2^n \plus{} n$.
1. We start with the given condition that \( 8^n + n \) is divisible by \( 2^n + n \). This can be written as:
   \[
   8^n + n \equiv 0 \pmod{2^n + n}
   \]
   which implies:
   \[
   8^n + n \equiv 0 \pmod{2^n + n}
   \]

2. Notice that \( 8^n = (2^n)^3 \). Therefore, we can rewrite the expression as:
   \[
   8^n + n = (2^n)^3 + n
   \]

3. To simplify the problem, we consider the expression modulo \( 2^n + n \):
   \[
   (2^n)^3 + n \equiv (-n)^3 + n \pmod{2^n + n}
   \]
   Simplifying further:
   \[
   (-n)^3 + n = -n^3 + n
   \]
   Thus, we need:
   \[
   -n^3 + n \equiv 0 \pmod{2^n + n}
   \]
   which simplifies to:
   \[
   2^n + n \mid n^3 - n
   \]

4. We now need to find all positive integers \( n \) such that \( 2^n + n \) divides \( n^3 - n \). For large \( n \), specifically \( n \geq 10 \), we observe that \( 2^n \) grows much faster than \( n^3 \). Hence, for \( n \geq 10 \):
   \[
   2^n > n^3
   \]
   This implies that \( 2^n + n \) is much larger than \( n^3 - n \), making it impossible for \( 2^n + n \) to divide \( n^3 - n \).

5. Therefore, we only need to check values of \( n \) from 1 to 9. We test each value:

   - For \( n = 1 \):
     \[
     2^1 + 1 = 3 \quad \text{and} \quad 1^3 - 1 = 0
     \]
     \( 3 \mid 0 \) is true.

   - For \( n = 2 \):
     \[
     2^2 + 2 = 6 \quad \text{and} \quad 2^3 - 2 = 6
     \]
     \( 6 \mid 6 \) is true.

   - For \( n = 3 \):
     \[
     2^3 + 3 = 11 \quad \text{and} \quad 3^3 - 3 = 24
     \]
     \( 11 \nmid 24 \) is false.

   - For \( n = 4 \):
     \[
     2^4 + 4 = 20 \quad \text{and} \quad 4^3 - 4 = 60
     \]
     \( 20 \mid 60 \) is true.

   - For \( n = 5 \):
     \[
     2^5 + 5 = 37 \quad \text{and} \quad 5^3 - 5 = 120
     \]
     \( 37 \nmid 120 \) is false.

   - For \( n = 6 \):
     \[
     2^6 + 6 = 70 \quad \text{and} \quad 6^3 - 6 = 210
     \]
     \( 70 \mid 210 \) is true.

   - For \( n = 7 \):
     \[
     2^7 + 7 = 135 \quad \text{and} \quad 7^3 - 7 = 336
     \]
     \( 135 \nmid 336 \) is false.

   - For \( n = 8 \):
     \[
     2^8 + 8 = 264 \quad \text{and} \quad 8^3 - 8 = 504
     \]
     \( 264 \nmid 504 \) is false.

   - For \( n = 9 \):
     \[
     2^9 + 9 = 521 \quad \text{and} \quad 9^3 - 9 = 720
     \]
     \( 521 \nmid 720 \) is false.

6. From the above calculations, the values of \( n \) that satisfy the condition are \( n = 1, 2, 4, \) and \( 6 \).

The final answer is \( \boxed{1, 2, 4, 6} \).