Determine the Euler totient function $\phi (n)$ of the following prime numbers:
(a) 13
(b) 211
(c) 311
(d) 1973
(e) 1999
(f) 2017
Let's remind ourselves of the Euler totient function. $\phi(n)$ counts the numbers from 1 to $n$ which are coprime to $n$,
For a prime $p$, all the numbers less than $p$ are coprime to $p$. In addition, a prime number is not coprime to itself. And so, for prime $p$, we have $\phi(p)=p-1$.
(a) $\phi(13) = 13-1 = 12$
(b) $\phi(211) = 211-1 = 210$
(c) $\phi(311) = 311-1 = 310$
(d) $\phi(1973) = 1973-1 = 1972$
(e) $\phi(1999)=1999-1 = 1998$
(f) $\phi(2017)=2017-1 =2016$