Gọi số cần tìm là [tex]\overline{ab}(a,b\in \mathbb{N},a\geq 1)[/tex]
Ta có: [tex]\left\{\begin{matrix} (10a+b)-(10b+a)=18\\ (10a+b)+(10b+a)^2=618 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} 9(a-b)=18\\ 10a+b+100b^2+20ab+a^2=618 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} a-b=2\\ a^2+(20b+10)a+100b^2+b=618 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} a=b+2\\ a^2+(20b+10)a+100b^2+b=618 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} a=b+2\\ (b+2)^2+(20b+10)(b+2)+100b^2+b=618 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} a=b+2\\ b^2+4b+4+20b^2+50b+20+100b^2+b-618=0 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} a=b+2\\ 121b^2+55b-594=0 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} a=b+2\\ 11(b-2)(11b+27)=0 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} a=b+2\\ b=2 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} a=4\\ b=2 \end{matrix}\right.\Rightarrow \overline{ab}=42[/tex]
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