Problem
NT-B1-M07-P012 When \(3^n\equiv1\)
#12
★★☆☆☆ Level 2 of 5
Find all positive \(n\) such that \(3^n\equiv1\pmod7\).
Use the order of \(3\) modulo \(7\).
From the previous problem, \(\operatorname{ord}_7(3)=6\). Therefore \(3^n\equiv1\pmod7\) if and only if \(6\mid n\). The answer is all positive multiples of \(6\).
Order turns an exponent condition into divisibility.