Totient divisibility along divisors ($d\mid n⟹\phi (d)\mid \phi (n)$)

Write $n=\prod p^{a_p}$ and $d=\prod p^{b_p}$, with $0\le b_p\le a_p$. The prime-power formula for the Euler totient function gives an integer contribution to $\phi (n)/\phi (d)$ from each prime: it is $p^{a_p-b_p}$ when $b_p>0$, and $p^{a_p-1}(p-1)$ when $b_p=0<a_p$. Multiplying proves the divisibility assertion, including $d=1$ with $\phi (1)=1$.