The Mysterious Number

Where $\color{#3D99F6}x$ represents the 5 digits $\color{#3D99F6}BCDEF$ .

$100000\color{#D61F06}A \color{#333333}+\color{#3D99F6}x \color{#333333}= \dfrac{10\color{#3D99F6}x \color{#333333}+\color{#D61F06}A\color{#333333}}{3}$

$300000\color{#D61F06}A \color{#333333}+3\color{#3D99F6}x \color{#333333}= 10\color{#3D99F6}x \color{#333333}+\color{#D61F06}A$

$299999\color{#D61F06}A \color{#333333} = 7\color{#3D99F6}x$

$42857\color{#D61F06}A \color{#333333} = \color{#3D99F6}x$

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The digit $\color{#D61F06}A$ cannot be greater than 2, otherwise $\color{#3D99F6}x$ ( $\color{#3D99F6}BCDEF$ ) would be a 6 digit number.

The digit $\color{#D61F06}A$ also cannot be $0$ , otherwise $\color{#D61F06}A\color{#3D99F6}BCDEF$ would be a 5 digit number.

This leaves $\color{#D61F06}A \color{#333333}= 1$ and $\color{#D61F06}A \color{#333333}= 2$ . Both of which actually satisfy this problem using different combinations of the same 6 distinct digits.

$\color{#D61F06}1\color{#3D99F6}42857\color{#333333} \times 3 = \color{#3D99F6}42857\color{#D61F06}1$

$\color{#D61F06}2\color{#3D99F6}85714\color{#333333} \times 3 = \color{#3D99F6}85714\color{#D61F06}2$

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The digits $1, 2, 4, 5 ,7, 8$ appear in all of the 6-digit numbers above.

$1+2+4+5+7+8 = \large\boxed{27}$

The number $ABCDEF$ is $100,000A+10,000B+1,000C+100D+10E+F$ , and $BCDEFA$ is $100,000B+10,000C+1,000D+100E+10F+A$ .

Since $BCDEFA$ is $3$ times bigger than $ABCDEF$ , $300,000A+30,000B+3,000C+300D+30E+3F=100,000B+10,000C+1,000D+100E+10F+A$ .

$299,999A=70,000B+7,000C+700D+70E+F, 42,857A=10,000B+1,000C+100D+10E+F$ (=the number $BCDEF$ ).

If $A=1$ , the number is $142857$ , and if $A=2$ , the number is $285714$ .

Thus, $A+B+C+D+E+F=1+4+2+8+5+7=2+8+5+7+1+4=27$ .