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20-11-1-mod-1000-




Question Number 198400 by cortano12 last updated on 19/Oct/23
  20^(11) −1 = ...(mod 1000)
$$\:\:\mathrm{20}^{\mathrm{11}} −\mathrm{1}\:=\:…\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$
Answered by MM42 last updated on 19/Oct/23
20^(11) −1≡^(1000) −1≡^(1000) 999
$$\mathrm{20}^{\mathrm{11}} −\mathrm{1}\overset{\mathrm{1000}} {\equiv}−\mathrm{1}\overset{\mathrm{1000}} {\equiv}\mathrm{999} \\ $$
Answered by mr W last updated on 19/Oct/23
20^(11) =2^(11) ×10^(11) =204800000000000  20^(11) −1=204799999999999  ...
$$\mathrm{20}^{\mathrm{11}} =\mathrm{2}^{\mathrm{11}} ×\mathrm{10}^{\mathrm{11}} =\mathrm{204800000000000} \\ $$$$\mathrm{20}^{\mathrm{11}} −\mathrm{1}=\mathrm{204799999999999} \\ $$$$… \\ $$
Answered by BaliramKumar last updated on 19/Oct/23
((20^(11) −1)/(1000)) = ((20^2 (20^3 )^3 −1)/(1000)) = ((20^2 (8000)^3 −1)/(1000))   ⇒ ((−1)/(1000)) = −1+1000 = 999
$$\frac{\mathrm{20}^{\mathrm{11}} −\mathrm{1}}{\mathrm{1000}}\:=\:\frac{\mathrm{20}^{\mathrm{2}} \left(\mathrm{20}^{\mathrm{3}} \right)^{\mathrm{3}} −\mathrm{1}}{\mathrm{1000}}\:=\:\frac{\mathrm{20}^{\mathrm{2}} \left(\mathrm{8000}\right)^{\mathrm{3}} −\mathrm{1}}{\mathrm{1000}}\: \\ $$$$\Rightarrow\:\frac{−\mathrm{1}}{\mathrm{1000}}\:=\:−\mathrm{1}+\mathrm{1000}\:=\:\mathrm{999}\: \\ $$

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