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Question Number 211632 by sonukgindia last updated on 15/Sep/24

Answered by A5T last updated on 15/Sep/24

$$\varphi \left(1000\right)=400$$$${2024}^{2024}\equiv 0\left(mod16\right);{2024}^{2024}\equiv 1\left(mod25\right)$$$$\Rightarrow {2024}^{2024}=25q+1\equiv \left(0mod16\right)\Rightarrow q\equiv 7\left(mod16\right)$$$$\Rightarrow {2024}^{2024}=25(16k+7)+1=400k+176\equiv 176\left(mod400\right)$$$$\Rightarrow {2024}^{{2024}^{2024}}\equiv {24}^{176}\left(mod1000\right)\equiv {\left({24}^{11}\right)}^{16}\equiv {24}^{16}$$$$\equiv {24}^{11}\times {24}^5\equiv {24}^6\equiv 976\left(mod1000\right)\Rightarrow ?=976$$

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