Is there a cleaner way to express the quotient $\frac{\langle(1,0),(0,1)\rangle}{\langle(2,-8)\rangle} \cong \frac{\mathbb{Z} \oplus \mathbb{Z}}{\langle(2,-8)\rangle}$?
To give the question some context, this is a presentation I got when trying to calculate a homology group here:
The quotient of the group $\mathbb Z^2$ by any infinite cyclic subgroup is isomorphic to the direct sum of $\mathbb Z$ and a finite cyclic group (which might be trivial). The order of the finite cyclic group is simply the greatest common divisor of the two components of the element that generates the kernel, in this case $gcd(2,-8)=2$. So your quotient is isomorphic to $\mathbb Z \oplus \mathbb Z / 2 \mathbb Z$.