Question Number 205922 by cortano12 last updated on 03/Apr/24
$$\:\:\:{x}\:=\:\mathrm{2}\:\left({mod}\:\mathrm{7}\right) \\ $$$$\:\:\:{x}=\mathrm{3}\:\left({mod}\:\mathrm{4}\right) \\ $$$$\:\:\:{x}=? \\ $$
Answered by Rasheed.Sindhi last updated on 03/Apr/24
$$\:\:\:{x}\:=\:\mathrm{2}\:\left({mod}\:\mathrm{7}\right)\:\wedge\:{x}=\mathrm{3}\:\left({mod}\:\mathrm{4}\right) \\ $$$${x}=\mathrm{7}{m}+\mathrm{2}\:\wedge\:\mathrm{7}{m}+\mathrm{2}\equiv\mathrm{3}\left({mod}\:\mathrm{4}\right. \\ $$$$\Rightarrow\mathrm{7}{m}\equiv\mathrm{1}\left({mod}\:\mathrm{4}\right. \\ $$$$\Rightarrow\mathrm{7}{m}\equiv\mathrm{1}+\mathrm{4}\centerdot\mathrm{5}=\mathrm{21}\left({mod}\:\mathrm{4}\right. \\ $$$$\Rightarrow{m}\equiv\mathrm{3} \\ $$$$\Rightarrow{x}=\mathrm{7}{m}+\mathrm{2}=\mathrm{7}\left(\mathrm{3}\right)+\mathrm{2}=\mathrm{23} \\ $$$$\Rightarrow{x}=\mathrm{23}+{lcm}\left(\mathrm{7},\mathrm{4}\right)×{k} \\ $$$$\Rightarrow{x}=\mathrm{23}+\mathrm{28}{k}\:\forall{k}\in\mathbb{Z} \\ $$
Answered by A5T last updated on 03/Apr/24
$${x}=\mathrm{7}{k}+\mathrm{2}\overset{\mathrm{4}} {\equiv}\mathrm{3}\Rightarrow\mathrm{3}{k}\overset{\mathrm{4}} {\equiv}\mathrm{9}\Rightarrow{k}\overset{\mathrm{4}} {\equiv}\mathrm{3}\Rightarrow{k}=\mathrm{4}{q}+\mathrm{3} \\ $$$$\Rightarrow{x}=\mathrm{7}\left(\mathrm{4}{q}+\mathrm{3}\right)+\mathrm{2}=\mathrm{28}{q}+\mathrm{23} \\ $$