Question Number 198418 by cortano12 last updated on 19/Oct/23
$$\:\mathrm{20}^{\mathrm{22}} −\mathrm{1}\:=\:…\:\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$$$ \\ $$
Answered by BaliramKumar last updated on 19/Oct/23
$$ \\ $$$$\frac{\mathrm{20}^{\mathrm{22}} −\mathrm{1}}{\mathrm{1000}}\:=\:\frac{\mathrm{20}\left(\mathrm{20}^{\mathrm{3}} \right)^{\mathrm{7}} −\mathrm{1}}{\mathrm{1000}}\:=\:\frac{\mathrm{20}\left(\mathrm{8000}\right)^{\mathrm{7}} −\mathrm{1}}{\mathrm{1000}}\: \\ $$$$\frac{\mathrm{20}\left(\mathrm{8}\cancel{\mathrm{000}}\right)^{\mathrm{7}} }{\mathrm{1}\cancel{\mathrm{000}}}\:−\:\frac{\mathrm{1}}{\mathrm{1000}}\:=\:\left(\mathrm{0}−\mathrm{1}\right)\:\mathrm{remainder} \\ $$$$\Rightarrow\:−\mathrm{1}+\mathrm{1000}\:=\:\mathrm{999}\: \\ $$