lean_workbook_plus_79821

So we want $2014^{2014^{2014}}\pmod {110}$ . Notice that $\varphi(110)=40$ so Euler's Theorem tells us that $2014^{2014^{2014}}\pmod {110}\equiv 2014^{2014^{2014}\pmod {40}}\pmod {110}$ . Again, $\varphi(40)=16$ so by Euler's Theorem $2014^{2014}\equiv 16\pmod {40}$ . Thus $2014^{2014^{2014}}\equiv 34^{16}\pmod {110}$ . Finally, once notices that $34^3=34\pmod {110}$ so that $34^{16}\equiv 34\cdot 34^5\equiv 34^6\equiv 34^2\equiv \boxed{56}\pmod {110}$ , which is our answer.