Q:

Difficulty with proof of statement ( addition mod n)

Proposition: Let $a,b,r\in \mathbb Z_n$ with n$>$2. Then, 
 $a+_nb =r$ (Which is remainder after a+b is divided by n) 
if and only if $\exists$ a $k \in \mathbb N$  such that  $a + b = kn +r $
I have to prove this, It seems really obvious from the definition of $+_n$ , So I do not know how to rigorously prove this.
(Note: I am unsure if Euclid's algorithm can be used in the proof as we have yet to cover that)

A:

If I've understood correctly your question, you're just saying that $a+b$ and $r$ are in the same coset mod $n$. That is, 
$
a+b-r \equiv 0 $ mod $n$, which of course is equivalent to the existence of an integer $k$ such that $a+b-r=kn$.

