Using the formula for $\displaystyle \varphi(n),$ compute $\displaystyle \varphi(27),\quad \varphi(81),$ and $\displaystyle \varphi(p^\alpha),$ where p is a prime number. Give a proof that the formula for $\displaystyle \varphi(n)$ is valid when $\displaystyle n=p^\alpha,$ where p is a prime number.

Here is what I have so far:

$\displaystyle

27=3^3 ==> \varphi(27)=27\left(1-\frac{1}{3}\right)=18

$

$\displaystyle

81=3^4 ==> \varphi(81)=81\left(1-\frac{1}{3}\right)=54

$

$\displaystyle

\varphi(p^\alpha)=p^\alpha\left(1-\frac{1}{p}\right)=p^\alpha-p^{\alpha-1}=p^{\alpha-1}(p-1).

$

I am trying to find a way to go about proving what it wants me to prove, but I can not see any methods that will work. Does anyone have any suggestions?