Order of the group of units of Z/nZ is Euler's totient

Statement

Let $n \geq 1$ be an integer. Then the order of the group $(\mathbb{Z}/n\mathbb{Z})^\times$ is $\varphi(n)$. Here $\varphi(n)$ is the number of integers $a$ with $0 \leq a \leq n - 1$ and $\gcd(a, n) = 1$.

Remarks

The proof is a count, because the units of Z/nZ are the residues coprime to n. For a prime $p$, the order is $\varphi(p) = p - 1$. For example, $(\mathbb{Z}/9\mathbb{Z})^\times = \{1, 2, 4, 5, 7, 8\}$ has order $\varphi(9) = 6$. With Lagrange's theorem, this gives Euler's theorem: $a^{\varphi(n)} \equiv 1 \pmod{n}$ for each $a$ coprime to $n$.

Explanations

L1 human

The size of the group of units modulo $n$ is the number of residues coprime to $n$. Euler's totient function counts exactly these residues. So the group has $\varphi(n)$ elements.