Proof of the following property about prime numbers Is there a proper proof of the following property: 

Let $p$ be a prime number. The number of invertible elements in $\mathbb{Z}/p^n\mathbb{Z}$ is $(p-1)p^{n-1}$.


Let $G=\mathbb{Z}/p^{n}\mathbb{Z}$. 
Lemma. For $a,n\in\mathbb{N}$, $ax\pmod{m}=1$ has a solution if and only if $\gcd(a,m)=1$.
Proof. 
$ab\pmod{m}=1\iff m\mid ab-1\iff\exists k\in\mathbb{N}$ such that $mk=ab-1\iff1=a(b)+m(-k)\iff\gcd(a,m)=1$.//
Now, by our lemma we know that the invertible elements in $G$ are precisely those that are relatively prime to $p^{n}$. If you are familiar with the Euler-totient function, then you know that the number of such elements is $\varphi(p^{n})=p^{n}-p^{n-1}=(p-1)p^{n-1}$. 
