$\overline{abc}+\overline{bca}=\left( a+b \right).100+\left( b+c \right).10+\left( a+c \right)=600$
$\left( a+b \right).100\vdots 100,\,\,600\vdots 100\Rightarrow \left( \left( b+c \right).10+\left( a+c \right) \right)\vdots 100$
$\left\{\begin{align} & b+c < 19 \\ & a+c < 19 \\ \end{align} \right.\Rightarrow \left( b+c \right).10+\left( a+c \right)=100$
$a+c < 19 \Rightarrow \left\{ \begin{align}& a+c=10 \\ & b+c=9 \\ \end{align} \right.$
$\Rightarrow a+b=5
\Rightarrow \left\{ \begin{align}& a+b=5 \\ & b+c=9 \\ & a+c=10 \\ \end{align} \right.$
$\Rightarrow \left\{ \begin{align}& a=3 \\ & b=2 \\ & c=7 \\ \end{align} \right.$
Vậy số cần tìm là 327.