Question Number 203490 by cortano12 last updated on 20/Jan/24
![1×3×5×7×9×...×2005 = ... (mod 1000)](https://www.tinkutara.com/question/Q203490.png)
$$\:\:\:\:\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}×\mathrm{9}×…×\mathrm{2005}\:=\:…\:\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$
Answered by AST last updated on 20/Jan/24
![x=1×3×5...×2005≡0(mod 125) x≡(1×3×5×7)^(250) ×1×3×5(mod 8)≡7(mod 8) x=125q≡7(mod 8)⇒5q≡15(mod 8)⇒q≡3(mod 8) 1×3×5...×2005=125(8k+3)≡375(mod 1000)](https://www.tinkutara.com/question/Q203493.png)
$${x}=\mathrm{1}×\mathrm{3}×\mathrm{5}…×\mathrm{2005}\equiv\mathrm{0}\left({mod}\:\mathrm{125}\right) \\ $$$${x}\equiv\left(\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}\right)^{\mathrm{250}} ×\mathrm{1}×\mathrm{3}×\mathrm{5}\left({mod}\:\mathrm{8}\right)\equiv\mathrm{7}\left({mod}\:\mathrm{8}\right) \\ $$$${x}=\mathrm{125}{q}\equiv\mathrm{7}\left({mod}\:\mathrm{8}\right)\Rightarrow\mathrm{5}{q}\equiv\mathrm{15}\left({mod}\:\mathrm{8}\right)\Rightarrow{q}\equiv\mathrm{3}\left({mod}\:\mathrm{8}\right) \\ $$$$\mathrm{1}×\mathrm{3}×\mathrm{5}…×\mathrm{2005}=\mathrm{125}\left(\mathrm{8}{k}+\mathrm{3}\right)\equiv\mathrm{375}\left({mod}\:\mathrm{1000}\right) \\ $$