Question Number 209892 by lmcp1203 last updated on 25/Jul/24
$${find}\:\:\mathrm{40}^{\mathrm{71}} {mod}\:\mathrm{437}.\:\:\:{thanks} \\ $$$${its}\:\mathrm{67}\:{but}\:{how}? \\ $$
Answered by Rasheed.Sindhi last updated on 25/Jul/24
$$\mathrm{40}^{\mathrm{25}} \equiv\mathrm{14}\:\left({mod}\:\mathrm{437}\right) \\ $$$$\left(\mathrm{40}^{\mathrm{25}} \right)^{\mathrm{3}} \equiv\mathrm{14}^{\mathrm{3}} \equiv\mathrm{122}\:\left({mod}\:\mathrm{437}\right) \\ $$$$\mathrm{40}^{\mathrm{75}} \equiv\mathrm{122}\:\left({mod}\:\mathrm{437}\right) \\ $$$$\begin{cases}{\mathrm{40}^{\mathrm{75}} \equiv\mathrm{122}\:\left({mod}\:\mathrm{437}\right)}\\{\mathrm{40}^{\mathrm{4}} \equiv\mathrm{54}\left({mod}\:\mathrm{437}\right)}\end{cases}\: \\ $$$$\begin{cases}{\mathrm{40}^{\mathrm{75}} \equiv\mathrm{122}+\mathrm{437}\left(\mathrm{8}\right)=\mathrm{3618}\:\left({mod}\:\mathrm{437}\right)…\left({i}\right)}\\{\mathrm{40}^{\mathrm{4}} \equiv\mathrm{54}\left({mod}\:\mathrm{437}\right)………\left({ii}\right)}\end{cases}\: \\ $$$$\left({i}\right)/\left({ii}\right):\:\mathrm{40}^{\mathrm{71}} \equiv\mathrm{67} \\ $$
Answered by lmcp1203 last updated on 25/Jul/24
$${thanks}\:{sir} \\ $$