Problem
NT-B2-M03-P012 An Equation for the Exponent
#12
★★★★☆ Level 4 of 5
Find all positive integers \(n\) such that \(5^n\equiv1\pmod{31}\).
Find the order of \(5\) modulo \(31\).
We have \(5^2=25\not\equiv1\pmod{31}\), while \(5^3=125\equiv1\pmod{31}\). Hence \( \operatorname{ord}_{31}(5)=3 \). Therefore \(5^n\equiv1\pmod{31}\) if and only if \(3\mid n\).
Show the connection between an exponent problem and the order criterion.