Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 5E / Solution

The Euler theorem states that if $\gcd (a,d)=1$, thenLet $r_1,\dots ,r_{\varphi (d)}$ be the reduced residue classes modulo $d$. Multiplication by $a$ permutes them, soTheir product is a unit and may be cancelled, proving the theorem.