A Weird Modular Arithmetic Question

Question

I saw this question somewhere and I was wondering if there's a nice closed form answer to it. It just seems like a troll question to me.

$2016^{2016} + 2018^{2016} (\bmod{2017}^{2016})$

I proceeded like this: $(2017-1)^{2016} + (2017+1)^{2016} (\mod 2017^{2016})$

then I expanded it using the binomial theorem and noticed that some parts cancel with each other and from there I got stuck.

I'm kind of new to number theory so I will highly appreciate a detailed explanation. Thank you!

Answer 1

Rewrite this expression as

$$2016^{2016}+2018^{2016}(\mod 2017^{2016})$$ $$\iff$$ $$(2017-1)^{2016} + (2017+1)^{2016} (\mod 2017^{2016}),$$ and then use the binomial theorem.