The Order Of $U(p)$ When $p$ Is Prime

If $p$ is prime, then every positive integer less than $p$ is relatively prime to $p$. This means that $U(p)=\{1,2,3,4,\ldots, p-1\}$, which means $|U(p)|=p-1$. The Euler $\varphi$ function is just the number of elements in $U(n)$, so we have $\varphi(p)=p-1$ for primes $p$.

If $n$ is not prime, then the set $U(n)$ does not contain every positive integer less than $n$, which makes it harder to work with $U(n)$ when $n$ is not prime.