\(2016^{2017}=\left(2^5.3^2.7\right)^{2017}=2^{10085}.3^{4034}.7^{2017}\)
Ta có: \(2^{20}\equiv100\left(mod100\right)\)\(\Rightarrow2^{10085}=\left(2^{20}\right)^{504}.2^5\equiv76.32\left(mod100\right)\equiv32\left(mod100\right)\)
\(3^{20}\equiv1\left(mod100\right)\)\(\Rightarrow3^{4034}=\left(3^{20}\right)^{201}.3^{14}\equiv1.\left(...69\right)\equiv69\left(mod100\right)\)
\(7^4\equiv1\left(mod100\right)\)\(\Rightarrow7^{2017}=\left(7^4\right)^{504}.7\equiv1.7\left(mod100\right)\equiv7\left(mod100\right)\)
Như vậy \(2016^{2017}\equiv32.69.7\left(mod100\right)\equiv56\left(mod100\right)\)
Vậy 2 chữ số tận cùng của 20162017 là 56
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