Question Number 202698 by Bambamamoudou last updated on 31/Dec/23
$${determine}\:{le}\:{reste}\:{de}\:{la}\:{division}\:{eucludienne}\:{de}\:\mathrm{2023}^{\mathrm{2019}} {par}\:\mathrm{13} \\ $$
Answered by Rasheed.Sindhi last updated on 01/Jan/24
$$\because\:{gcd}\left(\mathrm{2023},\mathrm{13}\right)=\mathrm{1} \\ $$$$\therefore\:\:\:\:\mathrm{2023}^{\phi\left(\mathrm{13}\right)} \equiv\mathrm{1}\left({mod}\:\mathrm{13}\right) \\ $$$$\:\:\:\:\:\:\:\:\mathrm{2023}^{\mathrm{12}} \equiv\mathrm{1}\left({mod}\:\mathrm{13}\right) \\ $$$$\left(\mathrm{2023}^{\mathrm{12}} \right)^{\mathrm{168}} \left(\mathrm{2023}\right)^{\mathrm{3}} \equiv\mathrm{2023}^{\mathrm{3}} \left({mod}\:\mathrm{13}\right) \\ $$$$\mathrm{2023}^{\mathrm{2019}} \equiv\left(\mathrm{13}×\mathrm{155}+\mathrm{8}\right)^{\mathrm{3}} \left({mod}\:\mathrm{13}\right) \\ $$$$\mathrm{2023}^{\mathrm{2019}} \equiv\left(\mathrm{8}\right)^{\mathrm{3}} =\mathrm{512}\equiv\mathrm{5}\left({mod}\:\mathrm{13}\right) \\ $$