# Multiplicative function evaluated at prime decompositions

Say $$f$$ is multiplicative and $$m = p^{\alpha}q^{\beta}$$ where $$p,q$$ are prime. Then do we have $$f(m) = f(p^{\alpha})f(q^{\beta})$$? If so why?. I know this hold when the powers are 1 but have not been given an explanation for higher powers.

• The standard definition of a multiplicative function says that $\gcd(a,b)=1\implies f(ab)=f(a)f(b)$. So, in your case, all you need is for the primes $p.q$ to be distinct.; – lulu May 10 at 10:38