2024/015) Find natual number n such that $2^n + n | 8^n + n$

 1,2,4,6 

We know that $x+ y | x^3+y^3$

Hence $2^n + n | (2^3)^n + n^3$

as $2^n + n | 8^n + n$

so $2^n + n | n^3-n$

so we must have $n^3-n =0$ or $2^n + n <  n^3-n$

$n^3-n= 0$ gives n = -1,0, 1 and out of theses only 1 is solution

we need to solve  $2^n + n <  n^3-n$ or  $2^n  <  n^3-2n$

let us find an upper bound for n putting a condition 

$2^n < n^3$ for  $n \lt 10$

putting n from 1 to 9 we see that $n \in \{1, 2,4,6\}$ satisfy the case and there is no other solution