Hint - Maybe this will help you move fwd
In this case x = 2009 is odd.
Had x been even life would have been simpler !
I have not tried this completely but would go down this route first - hope it works
Find the largest integer n, where 2009^n divides (2008^(2009^2010) + 2010^(2009 ^2008))
So, the only observation I've made is that you can rewrite this in the form:
(x-1)^(x^(x+1)) + (x+1)^(x^(x-1))
I also know that I have to use the binomial theorem, but I'm not exactly sure how that relates to finding the largest n possible...
If you can justify this isn't from an olympiad of some kind, please message me and I'll consider reopening this thread.
Until then, Thread Closed.
EDIT: The user has informed me that this comes from an olympiad competition that took place in 2008. As a result, feel free to make contributions to this thread.