Question Number 168538 by Mastermind last updated on 12/Apr/22
Answered by mr W last updated on 12/Apr/22
$${N}=\mathrm{7}{a}+\mathrm{3}=\mathrm{4}{b}+\mathrm{2} \\ $$$$\mathrm{4}{b}−\mathrm{7}{a}=\mathrm{1} \\ $$$$\Rightarrow{b}=\mathrm{7}{k}+\mathrm{2} \\ $$$$\Rightarrow{N}=\mathrm{4}\left(\mathrm{7}{k}+\mathrm{2}\right)+\mathrm{2}=\mathrm{28}{k}+\mathrm{10} \\ $$$${k}=\mathrm{0},\mathrm{1},…,\mathrm{35}\:\Rightarrow\mathrm{36}\:{numbers} \\ $$
Commented by Tawa11 last updated on 12/Apr/22
$$\mathrm{Great}\:\mathrm{sir}. \\ $$
Commented by Mastermind last updated on 13/Apr/22
$$ \\ $$$${Thanks}\:{man}'\:{love}\:{you}\:{guys} \\ $$
Commented by otchereabdullai@gmail.com last updated on 13/Apr/22
$$\mathrm{great} \\ $$
Answered by mathfreak last updated on 12/Apr/22
$${x}\:=\:\mathrm{3}\:{mod}\:\mathrm{7}\:=\:\mathrm{7}{a}\:+\:\mathrm{3} \\ $$$${x}\:=\:\mathrm{2}\:{mod}\:\mathrm{4}\:=\:\mathrm{4}{b}\:+\:\mathrm{2} \\ $$$$ \\ $$$$\mathrm{4}{x}\:=\:\mathrm{28}{a}\:+\:\mathrm{12} \\ $$$$\mathrm{7}{x}\:=\:\mathrm{28}{b}\:+\:\mathrm{14} \\ $$$$ \\ $$$$\mathrm{7}{x}\:−\:\mathrm{4}{x}\:=\:\mathrm{28}\left({a}−{b}\right)\:+\:\mathrm{2} \\ $$$$\mathrm{3}{x}\:=\:\mathrm{28}{c}\:+\:\mathrm{2} \\ $$$$\Rightarrow\:\mathrm{3}{x}\:=\:\mathrm{28}\left(\mathrm{3}{d}+\mathrm{1}\right)\:+\:\mathrm{2} \\ $$$$\mathrm{3}{x}\:=\:\mathrm{84}{d}\:+\:\mathrm{30} \\ $$$${x}\:=\:\mathrm{28}{d}\:+\:\mathrm{10} \\ $$$${x}\:=\:\left\{\mathrm{28}\left(\mathrm{0}\right)+\mathrm{10}\:,\:\mathrm{28}\left(\mathrm{1}\right)+\mathrm{10}\:,…,\mathrm{28}\left(\mathrm{35}\right)+\mathrm{10}\right\} \\ $$$${n}\left({x}\right)_{\mathrm{0}<{x}<\mathrm{1000}} \:=\:\mathrm{35}\:+\:\mathrm{1}\:=\:\mathrm{36} \\ $$$$ \\ $$
Commented by Tawa11 last updated on 12/Apr/22
$$\mathrm{Great}\:\mathrm{sir}. \\ $$
Commented by Mastermind last updated on 13/Apr/22
$${Thanks},\:{you}'{re}\:{extremely}\:{did}\:{a}\:{great}\:{job} \\ $$$$ \\ $$