[tex]\it \left.\begin{aligned}n:4=a\ \ rest\ 2 \Rightarrow\ n=4a+2|_{+2} \Rightarrow\ n+2=4a+4 \Rightarrow\ n+2\in M_4\\ \\ n:5=b\ \ rest\ 3 \Rightarrow\ n=5b+3|_{+2} \Rightarrow\ n+2=5b+5 \Rightarrow\ n+2\in M_5\\ \\ n:6=c\ \ rest\ 4 \Rightarrow\ n=6c+4|_{+2} \Rightarrow\ n+2=6c+6 \Rightarrow\ n+2\in M_6\\ \\ n:7=d\ \ rest\ 5 \Rightarrow\ n=7d+5|_{+2} \Rightarrow\ n+2=7d+7 \Rightarrow\ n+2\in M_7\end{aligned}\right\}\Rightarrow\\ \\ \\ \Rightarrow n+2\in M_4\cap M_5\cap M_6\cap M_7[/tex]
[tex]\it [4,\ 5,\ 6,\ 7]=420 \Rightarrow n+2=420k|_{-2} \Rightarrow n\in 420k-2,\ k\in\mathbb{N}\\ \\ 200<n<2000 \Rightarrow 200<420k-2<2000\\ \\ k=1 \Rightarrow n=420-2=418\\ \\ k=2 \Rightarrow n=420\cdot2-2=838\\ \\ k=3 \Rightarrow n=420\cdot3-2=1258\\ \\ k=4 \Rightarrow n=420\cdot4-2=1678\\ \\ k=5 \Rightarrow n=420\cdot5-2=2098\ (nu\ \ convine)[/tex]
Numerele cerute sunt: 418, 838, 1258 și 1678
