Show that following modules are isomorphic 
I have $M=\mathbb{Z^2/\langle(-2,5)\rangle}$ as $\mathbb{Z}$-module and have to show that it is isomorphic with $\mathbb{Z}$.

I have found an isomorphism $f: \mathbb{Z^2}\rightarrow\mathbb{Z^2}$ which sends $(-2,5)$ to $(0,1)$ and $(1,-2)$ to $(1,0)$. Can I say that $f$ induces an isomorphism $\mathbb{Z^2}/\langle(-2,5)\rangle\rightarrow\mathbb{Z^2}/\langle(0,1)\rangle$ and then conclude that it is isomorphic to $\mathbb{Z}$?
I could say that if $\langle(-2,5)\rangle$ is the kern of $f$ and $\mathbb{Z^2}/\langle(0,1)\rangle$ the image. Why is this the case, and how can I then conclude that it is isomorphic to $\mathbb{Z}$?

If $f:G\to G$ is a module automorphism and $H$ is a submodule of $G$, then $G/H\simeq G/f(H)$.
In your case the automorphism is $f(a,b)=(5a+2b,2a+b)$ and then $$\mathbb Z^2/\langle(-2,5)\rangle\simeq \mathbb Z^2/\langle(0,1)\rangle.$$
But $\langle(0,1)\rangle=\{0\}\times\mathbb Z$. Now $$\mathbb Z^2/\{0\}\times\mathbb Z\simeq\mathbb Z/\{0\}\times\mathbb Z/\mathbb Z\simeq\mathbb Z.$$
