Problem
NT-B1-M07-P016 Remainder Modulo \(28\)
#16
★★★☆☆ Level 3 of 5
Find the remainder of \(3^{2026}\) modulo \(28\).
Split the modulus \(28\) into \(4\) and \(7\).
Modulo \(4\), \(3^{2026}\equiv1\), since the exponent is even. Modulo \(7\), \(3^6\equiv1\), and \(2026\equiv4\pmod6\), so \(3^{2026}\equiv3^4=81\equiv4\pmod7\). We need a number congruent to \(1\) modulo \(4\) and to \(4\) modulo \(7\). It is \(25\). The answer is \(25\).
CRT is often more convenient than direct work modulo a composite number.