Question Number 211632 by sonukgindia last updated on 15/Sep/24
Answered by A5T last updated on 15/Sep/24
$$\phi\left(\mathrm{1000}\right)=\mathrm{400} \\ $$$$\mathrm{2024}^{\mathrm{2024}} \equiv\mathrm{0}\left({mod}\:\mathrm{16}\right);\mathrm{2024}^{\mathrm{2024}} \equiv\mathrm{1}\left({mod}\:\mathrm{25}\right) \\ $$$$\Rightarrow\mathrm{2024}^{\mathrm{2024}} =\mathrm{25}{q}+\mathrm{1}\equiv\left(\mathrm{0}\:{mod}\:\mathrm{16}\right)\Rightarrow{q}\equiv\mathrm{7}\left({mod}\mathrm{16}\right) \\ $$$$\Rightarrow\mathrm{2024}^{\mathrm{2024}} =\mathrm{25}\left(\mathrm{16}{k}+\mathrm{7}\right)+\mathrm{1}=\mathrm{400}{k}+\mathrm{176}\equiv\mathrm{176}\left({mod}\:\mathrm{400}\right) \\ $$$$\Rightarrow\mathrm{2024}^{\mathrm{2024}^{\mathrm{2024}} } \equiv\mathrm{24}^{\mathrm{176}} \left({mod}\:\mathrm{1000}\right)\equiv\left(\mathrm{24}^{\mathrm{11}} \right)^{\mathrm{16}} \equiv\mathrm{24}^{\mathrm{16}} \\ $$$$\equiv\mathrm{24}^{\mathrm{11}} ×\mathrm{24}^{\mathrm{5}} \equiv\mathrm{24}^{\mathrm{6}} \equiv\mathrm{976}\left({mod}\:\mathrm{1000}\right)\Rightarrow?=\mathrm{976} \\ $$