[ Euler's function ]

Euler's function $\varphi(n)$ is defined as the number of integers from 1 to $n$ that are coprime with $n$. Find

a) $\varphi(17) ;$ b) $\varphi(p) ;$ c) $\varphi\left(p^{2}\right) ;$ d) $\varphi\left(p^{\alpha}\right)$.
г) When calculating $\varphi\left(p^{\alpha}\right)$, we need to exclude all numbers divisible by $p$. Among the numbers from 1 to $p^{\alpha}$, there are exactly $p^{\alpha-1}$ such numbers.

Therefore, $\varphi\left(p^{\alpha}\right)=p^{\alpha-1}(p-1)$.

## Answer

a) 16;
b) $p-1$;
c) $p(p-1)$;
d) $p^{\alpha-1}(p-1)$.