I'm not sure I understand your post.
Is this what you mean?
For what value of is this expression divisible by ?
Note that , which is divisible by .
So, for what value of n are , , , and equal to , , and , respectively?
The answer to this is of course . If you take anything to the power of , it will be equal to itself.
If you need to find all values of for which this expression is divisible by , note that every odd integer gives a number that is divisible by .