Totient and division I know that if the arguments are relatively prime, I can write
$$
\phi(ab)=\phi(a)\phi(b)
$$
can I conclude that if $b\mid a$
$$
\frac{\phi(a)}{\phi(b)}=\phi(\frac{a}{b})
$$
Thanks

No.$\let\phi\varphi$ The assertion $\frac{\phi(a)}{\phi(b)}=\phi(\tfrac ab)$ is equivalent to complete multiplicativeness: Let $a=bk$, then it says
$$\phi(bk)=\phi(b)\phi(k)\qquad\text{for arbitrary }b,k$$
which would mean that $\phi$ is completely multiplicative, which it is not. It is only true if $\gcd(b,k)=1$.
