\[\begin{align}
& \overline{abc}-2(a-2b+4c)=100a+10b+c-2a+4b-8c=98a+14b-7c\vdots 7 \\
& 2(a-2b+4c)\vdots 7\Rightarrow \overline{abc}\vdots 7(1) \\
& \overline{abc}-(a-2b+4c)=100a+10b+c-a+2b-4c=99a+12b-3c\vdots 3 \\
& a-2b+4c\Rightarrow \overline{abc}\vdots 3(2) \\
& (3;7)=1(3) \\
& (1);(2);(3)\Rightarrow \overline{abc}\vdots 21 \\
\end{align}\]