Problem
NT-B1-M07-P006 A Power of Five Modulo \(11\)
#6
★★☆☆☆ Level 2 of 5
Find the remainder of \(5^{2026}\) modulo \(11\).
By Fermat, reduce the exponent modulo \(10\).
Since \(11\) is prime, \(5^{10}\equiv1\pmod{11}\). Also \(2026\equiv6\pmod{10}\), so \(5^{2026}\equiv5^6\). Now \(5^2\equiv3\), \(5^4\equiv9\), hence \(5^6\equiv9\cdot3=27\equiv5\pmod{11}\).
The exponent is reduced, but the remaining residue still must be computed.