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Exercise (The Subgroups Of $\mathbb{Z}$ are $n\mathbb{Z}$)
We know that $n\mathbb{Z}$ is a subgroup of $\mathbb{Z}$ for every integer $n$. Show that these are the only subgroups of $\mathbb{Z}$. In particular this means that the span of $k$ integers, which is a subgroup of $\mathbb{Z}$, must be equal to $d\mathbb{Z}$ for some $d\in \mathbb{Z}$.
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The integers are a cyclic group, so every subgroup is also cyclic. If we let $H$ be a subgroup of $\mathbb{Z}$, then we know there exists $d\in\mathbb{Z}$ such that $H=\left<d\right>$. This shows that $H$ equals the set of multiples of $d$, which means that $H=d\mathbb{Z}$.