[tex]\it \{a+b,\ b+c,\ c+a\}\ i.\ p.\ \{2,\ 3,\ 4\}\ \Rightarrow 2(a+b)=3(b+c)=4(c+a) \Rightarrow \\ \\ \Rightarrow 2a+2b=3b+3c=4c+4a\\ \\ 2a+2b=4c+4a \Rightarrow 2b-4c=4a-2a\Rightarrow 2b-4c=2a|_{:2}\Rightarrow a=b-2c\ \ \ (1)\\ \\ 3b+3c=4c+4a\Rightarrow 3b=4c+4a-3c\stackrel{(1)}{\Longrightarrow}\ 3b=4c+4b-8c-3c\Rightarrow \\ \\ \Rightarrow 3b=4b-7c\Rightarrow 7c=4b-3b\Rightarrow 7c=b\Rightarrow b=7c\ \ \ (2)\\ \\ (1),\ (2)\Rightarrow a=7c-2c\Rightarrow a=5c\ \ \ (3)[/tex]
[tex]\it \overline{abc}=100a+10b+c\ \stackrel{(2),(3)}{\Longrightarrow}\ \overline{abc}=500c+70c+c=571c \Rightarrow c=1\\ \\ Deci,\ \overline{abc}=571[/tex]
