2021/026) Find positive integer n for which $2^n + n | 8^n + n$

We have $8^n + n ^3 = (2^n)^3 + n^3$

So

$2^n + n | 8^n + n^3$

using above and the given condition

$2^n + n |  n^3-n$

Now there are 2 cases

1) $n^3-n = 0$ which gives n = 1 ( as other roots n = -1 and 0 are not positive

2) or $n^3-n > 0$ as for positive integer n >=2 ( n= 1 as already considered) 

as $2^n + n | n^3-n$ so we have $n^3- n \ge 2^n + n$

or $n^3 \ge 2^n + 2n$

we can show by induction that for $n > 10 $ $2^n > n^3$ 

so $2 \le n \ le 9$

by checking out the values from 2 to 9 we get

$ n \in \{2,4,6\}$

so we ahve the solution set

$n \in \{ 1,2,4,6\}$