Question Number 203354 by SEKRET last updated on 17/Jan/24
$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{8}^{\mathrm{2022}} =\:\:\boldsymbol{\mathrm{a}}\:\:\boldsymbol{\mathrm{mod}}\left(\mathrm{100}\right)\:\:\:\:\:\:\boldsymbol{\mathrm{a}}=? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Answered by Rasheed.Sindhi last updated on 17/Jan/24
$${Can}\:{be}\:{observed}\:{after}\:{some}\: \\ $$$${experiments}\:{that}: \\ $$$$\mathrm{8}^{{r}+\mathrm{20}{q}} \equiv\mathrm{8}^{{r}} \left({mod}\:\mathrm{100}\right) \\ $$$${r}+\mathrm{20}{q}=\mathrm{2022}=\mathrm{2}+\mathrm{20}×\mathrm{100} \\ $$$$\mathrm{8}^{\mathrm{2022}} =\mathrm{8}^{\mathrm{2}+\mathrm{20}×\mathrm{100}} \equiv\mathrm{8}^{\mathrm{2}} =\mathrm{64}\left({mod}\:\mathrm{100}\right) \\ $$
Commented by SEKRET last updated on 17/Jan/24
$$ \\ $$Thank you sir. I will wait for other solutions