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 $\map \phi n$ denote the Euler $\phi$ function of $n$. Let $p$ be a divisor of $n$.

Then $p - 1$ is a divisor of $\map \phi n$.

Proof
From Euler Phi Function of Integer:


 * $\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$

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.