Question Number 203490 by cortano12 last updated on 20/Jan/24
$$\:\:\:\:\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}×\mathrm{9}×...×\mathrm{2005}\:=\:...\:\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$
Answered by deleteduser1 last updated on 20/Jan/24
$${x}=\mathrm{1}×\mathrm{3}×\mathrm{5}...×\mathrm{2005}\equiv\mathrm{0}\left({mod}\:\mathrm{125}\right) \\ $$$${x}\equiv\left(\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}\right)^{\mathrm{250}} ×\mathrm{1}×\mathrm{3}×\mathrm{5}\left({mod}\:\mathrm{8}\right)\equiv\mathrm{7}\left({mod}\:\mathrm{8}\right) \\ $$$${x}=\mathrm{125}{q}\equiv\mathrm{7}\left({mod}\:\mathrm{8}\right)\Rightarrow\mathrm{5}{q}\equiv\mathrm{15}\left({mod}\:\mathrm{8}\right)\Rightarrow{q}\equiv\mathrm{3}\left({mod}\:\mathrm{8}\right) \\ $$$$\mathrm{1}×\mathrm{3}×\mathrm{5}...×\mathrm{2005}=\mathrm{125}\left(\mathrm{8}{k}+\mathrm{3}\right)\equiv\mathrm{375}\left({mod}\:\mathrm{1000}\right) \\ $$