No finitely generated abelian group is divisible

Prove that no finitely generated abelian group is divisible.

Let $A$ be a nontrivial divisible finitely generated abelian group. By FTFGAG, we have $A \cong \mathbb{Z}^r \times B$, where $B$ is a finite abelian group. By this previous exercise, $A$ is divisible if and only if $\mathbb{Z}^r$ and $B$ are divisible. By this previous exercise, if $B$ is nontrivial then it is not divisible; thus $B = 1$. Again by this result, since $r \geq 1$, $A$ is divisible if and only if $\mathbb{Z}$ is divisible.

Note that there does not exist an element $k \in \mathbb{Z}$ such that $2k = 1$. Thus $\mathbb{Z}$ is not divisible, hence $A$ is not divisible, a contradiction. Thus no such group $A$ exists.