$(\mathbb{Z}/12)^\times=\{[1],[5],[7],[11]\}$ and $\varphi(12)=4$

Given: The quotient $\mathbb Z/12$ and its unit group.

The classes modulo $12$ have the unique standard representatives $0,\ldots,11$ (For $n\ge 1$, every class in $\mathbb{Z}/n$ has one representative $r$ with $0\le r<n$, so $\lvert\mathbb{Z}/n\rvert=n$; while $\mathbb{Z}/0$ is in bijection with $\mathbb{Z}$).

The class $[a]_{12}$ is a unit exactly when $\gcd(a,12)=1$, and $\varphi(12)$ counts these units (For $n\ge1$, $[a]_n$ is a unit if and only if $\gcd(a,n)=1$, The unit group $(\mathbb{Z}/n)^\times$ and Euler's totient $\varphi(n)=\lvert(\mathbb{Z}/n)^\times\rvert$ for $n\ge1$).

Euler's finite prime-divisor product gives $\varphi(n)=\prod_{p^k\parallel n}(p^k-p^{k-1})$ (Euler's product formula $\varphi(n)=n\prod_{p\mid n}(1-1/p)=\prod_{p^k\parallel n}(p^k-p^{k-1})$ for $n\ge1$, stated through a finite injective list of its prime divisors).

Equality of residue classes is congruence of representatives (The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$), and congruence means divisibility of their difference (Congruence modulo an integer: $a\equiv b\pmod n$ when $n\mid(a-b)$, including the moduli $0$ and $1$).

Products of residue classes are computed by multiplying representatives: $[a]_{12}[b]_{12}=[ab]_{12}$ (Addition and multiplication on $\mathbb{Z}/n$ by $[a]_n+[b]_n=[a+b]_n$ and $[a]_n[b]_n=[ab]_n$).