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