Past exam of the options II course of the University of Oxford / 2005 / C / 2 / a / i / Solution

Euler's totient function $\varphi (n)$ counts the integers $1\le a\le n$ coprime to $n$, equivalently the invertible residue classes modulo $n$. Among $1,\dots ,p^e$, exactly the $p^{e-1}$ multiples of $p$ are not coprime to the prime power. HenceFactor $n={\prod }_p{p}^{e_p}$. Its distinct prime-power factors are pairwise coprime, so the permitted multiplicativity givesFor $n=1$, the empty product is one and agrees with $\varphi (1)=1$.