Euler Phi Function of Integer/Corollary

Corollary to Euler Phi Function of Integer
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $p$ be a prime number.

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

Then $p - 1$ is a divisor of $\phi \left({n}\right)$.

Proof
From Euler Phi Function of Integer:


 * $\displaystyle \phi \left({n}\right) = n \prod_{p \mathrel \backslash n} \left({1 - \frac 1 p}\right)$

Let $n$ be expressed as its prime decomposition:


 * $n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}, p_1 < p_2 < \ldots < p_r$

Then:

Hence the result.