Möbius Function is Multiplicative/Corollary

Theorem
The Möbius function $\mu$ is a multiplicative function:
 * $m \perp n \implies \mu \left({m n}\right) = \mu \left({m}\right) \mu \left({n}\right)$

where $m, n \in \Z_{>0}$.

Corollary
Let $\gcd \left\{{m, n}\right\} > 1$.

Then:
 * $\mu \left({m n}\right) = 0$

where $\mu$ denotes the Möbius function.

Proof
Let $\gcd \left\{{m, n}\right\} = k$ where $k > 1$.

Then $m = k r$ and $n = k s$ for some $r, s \in \Z$.

Thus $m n = k^2 r s$.

From Integer Expressible as Product of Primes there exists $p \in \Z$ such that $p$ is prime and $p \mathop \backslash k$.

That is:
 * $\exists t \in \Z: k = p t$

and so:
 * $m n = p^2 t^2 r s$

That is:
 * $p^2 \mathop \backslash m $

Hence the result by definition of the Möbius function.