Problem
ALG-B3-M08-P015 Linear Involution Modulo a Prime
#15
★★★★★ Level 5 of 5
Let \(p\) be an odd prime, \(f(x)=ax\) on \(\mathbb Z/p\mathbb Z\), and \(f^2(x)=x\) for all \(x\). Find all possible \(a\).
The condition means \(a^2x\equiv x\).
For all \(x\), \(a^2x\equiv x\pmod p\), so \(a^2\equiv1\pmod p\). Thus \((a-1)(a+1)\equiv0\pmod p\). Since \(p\) is prime, \(a\equiv1\) or \(a\equiv-1\pmod p\).
Modular iteration of a coefficient.