Problem
NT-B2-M02-P006 Square Roots of Minus One
#6
★★★☆☆ Level 3 of 5
Solve the congruence \(x^2\equiv-1\pmod{13}\).
We need \(x^2\equiv12\pmod{13}\). Check residues up to \(6\).
The squares \(0^2,1^2,\ldots,6^2\) modulo \(13\) are \(0,1,4,9,3,12,10\). Hence \(x\equiv5\) or \(x\equiv-5\equiv8\pmod{13}\).
Remind students that it is enough to check pairs \(x\) and \(-x\).