Question Number 197752 by cortano12 last updated on 27/Sep/23
$$\:\mathrm{find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{m} \\ $$$$\:\mathrm{such}\:\mathrm{that}\:\mathrm{m}^{\mathrm{19}} =\:\mathrm{1800}\:\left(\mathrm{mod}\:\mathrm{2029}\right) \\ $$
Answered by AST last updated on 27/Sep/23
$${m}^{\mathrm{2028}} =\left({m}^{\mathrm{19}} \right)^{\mathrm{107}} {m}^{−\mathrm{5}} \equiv\mathrm{1}\left({mod}\:\mathrm{2029}\right) \\ $$$$\Rightarrow{m}^{\mathrm{5}} \equiv\left(−\mathrm{229}\right)^{\mathrm{107}} \Rightarrow{m}^{\mathrm{5}} \equiv\mathrm{1183}\left({mod}\:\mathrm{2029}\right) \\ $$$${m}^{\mathrm{19}} =\frac{\left({m}^{\mathrm{5}} \right)^{\mathrm{4}} }{{m}}\equiv\mathrm{1800}\left({mod}\:\mathrm{2029}\right) \\ $$$$\Rightarrow\mathrm{1800}{m}\equiv\left(\mathrm{1183}\right)^{\mathrm{4}} \equiv\mathrm{1584}\left({mod}\:\mathrm{2029}\right) \\ $$$$\Rightarrow\mathrm{25}{m}\equiv\mathrm{22}\left({mod}\:\mathrm{2029}\right) \\ $$$$\Rightarrow\mathrm{25}{m}\equiv\mathrm{22}+\mathrm{2029}×\mathrm{7}=\mathrm{2425}=\mathrm{14225} \\ $$$$\Rightarrow{m}\equiv\mathrm{569}\left({mod}\:\mathrm{2029}\right) \\ $$$$\Rightarrow{min}\left\{{m}\right\}=\mathrm{569} \\ $$