Problem - 2508
Let $p$ be a prime number and $n$ be a positive integer. Show that $\varphi(p^n)=p^n - p^{n-1}$ where $\varphi(n)$ is the Euler's totient function.
By the definition, $$\varphi(p^n) = p^n\left(1-\frac{1}{p}\right)=p^n-p^{n-1}$$
By the definition, $$\varphi(p^n) = p^n\left(1-\frac{1}{p}\right)=p^n-p^{n-1}$$