Euler Phi Function of Product with Prime/Corollary

Theorem
Let $n \in \Z_{>0}$ be a positive integer.

Let $\phi \left({n}\right)$ denote the Euler $\phi$ function of $n$.

Then:
 * $d \mathop \backslash n \implies \phi \left({d}\right) \mathop \backslash \phi \left({n}\right)$

where $d \mathop \backslash n$ denotes that $d$ is a divisor of $n$.

Proof
Let $d \mathop \backslash n$.

We can write $n$ as $n = d p_1 p_2 p_3 \cdots p_r$, where $p_1, p_2, \ldots, p_r$ are all the primes (not necessarily distinct) which divide $\dfrac n d$.

Thus, repeatedly using Euler Phi Function of Product with Prime:

As the last expression is $\phi \left({n}\right)$, the result follows from Divides is Partial Ordering on Positive Integers.