5. A six-digit natural number has its first digit (the ten-thousands place) as 2. If this digit is moved from the first to the last position, the newly formed number will be three times the original number. Which numbers are these?

Each task is scored out of 10 points.

The use of a pocket calculator or any reference materials is not allowed.
First method:

Let $\overline{2 a b c d e}$ be the desired initial number. Then the newly obtained number is $\overline{a b c d e 2}$. 2 POINTS

It follows that $\overline{a b c d e 2}=3 \cdot \overline{2 a b c d e}$.

Further, we have:

```
\(10 \cdot \overline{a b c d e}+2=3 \cdot(200000+\overline{a b c d e})\)
2 POINTS
```

$10 \cdot \overline{a b c d e}+2=600000+3 \cdot \overline{a b c d e}$

1 POINT

$7 \cdot \overline{a b c d e}=600000-2$

1 POINT

$\overline{a b c d e}=599998: 7$

1 POINT

$\overline{a b c d e}=85714$

1 POINT

The initial number is 285714, and the newly obtained number is 857142.

1 POINT

TOTAL 10 POINTS