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20-22-1-mod-1000-




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

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