I doubt anyone wants to read closely a long proof of something that is quite simple (no offense).

I would suggest learning modular arithmetic if you wanted to clarify your understanding of integers. Almost every book about elementary number theory begins with those ideas.

Here is a very simple proof :

is a field. In perticular its nonzero elements form a group under the multiplication of . But in any finite group of order , we have for all . Hence in this case , or in other words is divisible by .