Completely Additive Function is Additive

Theorem
Let $f: \Z \to \Z$ be a completely additive function on the ring of integers $\Z$.

Then $f$ is also additive.

Proof
Let $m, n$ be coprime integers.

Then in particular, $m, n \in \Z$.

Hence, since $f$ is completely additive:


 * $f \left({m \times n}\right) = f \left({m}\right) + f \left({n}\right)$

and $f$ is additive, as desired.