Identifying Abelian group

Question

I was asked by some one to identify quotient group $\frac{\mathbb{Z} \oplus \mathbb{Z}}{\langle (1,2),(3,4) \rangle}$ and I argued that since $(3,4)-2(1,2)=(1,0)$ this implies $(m,0) \in \langle (1,2),(3,4) \rangle \forall m \in \mathbb{Z}$ . Also $(0,2) =3(1,2)-(3,4)$ and this says $ (0,2m) \in \langle (1,2),(3,4) \rangle \forall m \in \mathbb{Z}$. Thus for any arbitrary element $(m,n) + \langle (1,2),(3,4) \rangle $ in the quotient group it will be written as $(m,0)+(0,n) + \langle (1,2),(3,4) \rangle = (0,n)+ \langle (1,2),(3,4) \rangle $ where the only choices for n are 0 or 1. Thus this quotient only contains two elements namely $ (0,0)+ \langle (1,2),(3,4) \rangle $ and $(0,1) + \langle (1,2),(3,4) \rangle $ , Hence isomorphic to $\mathbb{Z_{2}}$. Now i want to ask whether my arguments are correct and is there any other method to identify this type of quotient? Please Verify.