Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 324 / 1 / b / i / Solution

Since $\gcd (x,N)=1$, multiplication by $x$ modulo $N$ is a bijection of ${\mathbb{Z}}_N$, with inverse multiplication by the modular inverse $x^{-1}$. Hence $U_x$ permutes the standard orthonormal basis. It preserves all inner products and satisfiesso $U_x$ is a unitary operator.