4. Find all natural $n$ such that the number $8^{n}+n$ is divisible by $2^{n}+n$.

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The provided text has been translated from Russian to English, maintaining the original formatting and structure.
Answer: $n=1,2,4,6$.

By the formula of abbreviated multiplication, the number $8^{n}+n^{3}$ is divisible by $2^{n}+n$, so the condition of the problem is equivalent to the condition $n^{3}-n \vdots 2^{n}+n$. But for $n \geqslant 10$ the inequality $n^{3}<2^{n}$ holds. It remains to check 9 options and get the answer.

+ For using the inequality $n^{3}<2^{n}$ for $n \geqslant 10$ without proof, points are not deducted.

$\pm$ The correct approach to the solution, but answers are lost. Deduct one point for each lost answer.

干 It is established that $n^{3}-n \vdots 2^{n}+n$ or a similar divisibility, where the dividend is usually less than the divisor. 3 points

- All answers are found without justification. 1 point