Problem
NT-B2-M03-P013 When a Power Equals \(-1\)
#13
★★★★☆ Level 4 of 5
Find all positive integers \(n\) such that \(2^n\equiv-1\pmod{17}\).
Notice that \(2^4\equiv-1\pmod{17}\).
Compute \(2^4=16\equiv-1\pmod{17}\), hence \(2^8\equiv1\). The order of \(2\) modulo \(17\) is \(8\), because the smaller divisors \(1,2,4\) do not give \(1\). Thus \(2^n\equiv-1\) exactly when \(n\equiv4\pmod8\).
It is important not to write \(n\equiv4\) without specifying the order modulus.