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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.

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    $\begingroup$ 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.; $\endgroup$ – lulu May 10 at 10:38

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