Equality of rank for homology and cohomology groups via the universal coefficient theorem
Solution 1:
Every f.g. abelian group is the direct sum of finitely many finite cyclic groups and a f.g. torsion-free (hence free) abelian group. $\mathrm{Ext}_\mathbb{Z}^1 (-, \mathbb{Z})$ preserves finite direct sums, so it suffices to show that $\mathrm{Ext}_\mathbb{Z}^1 (M, \mathbb{Z})$ is a finite group if $M$ is a finite cyclic group.
In fact, for all positive integers $m$, $$\mathrm{Ext}_\mathbb{Z}^1 (\mathbb{Z} / m \mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z} / m \mathbb{Z}$$ as may be verified by direct calculation. This proves the claim.