Find all six-digit numbers that are reduced to a third when the last digit is moved to the first place.

#
Let $\overline{a b c d e f}=3 \cdot \overline{f a b c d e}$. Consider the number $\alpha$, which can be expressed as a repeating decimal with the period $\overline{\overline{b b c d e f}}: \alpha=0,(f a b c d e)=1 / 3 \cdot 0,(a b c d e f)$. Then $10 \alpha=f,(a b c d e f)=f+3 \alpha$. From this, $\alpha=f / 7$. It remains to try different values of $f$.

## Answer

428571 and 857142.