Isomorphisms $\mathbb{Z}/n \mathbb{Z}\cong \mathbb{Z}_n$ and $K[x]/(f)\cong K[x]\bmod f$ 
Let $n\in \mathbb{Z}^+$. How do I prove that $\mathbb{Z}/n\mathbb{Z}$ is isomorphic to $\mathbb{Z}_n$? 

Is there any good homomorphism $\phi$ I could use that graphs  $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}_n$, that is one to one and onto. I'm not sure how to come up with one. 
Thank you for any input! 

Hint: Given an integer $m$, use the division algorithm to write $$m = nq + r$$ Define $\phi(m) = r \in \mathbb{Z}_n$ and check the conditions for the first isomorphism theorem (or just show that $\phi$ works directly).
