Problem 7.1. (15 points) Find the smallest ten-digit natural number, all digits of which are different, such that when all even digits are erased, 97531 remains, and when all odd digits are erased, 02468 remains.
Answer: 9024675318.

Solution. Let $A=9024675318$. Consider an arbitrary number $N=\overline{a_{1} a_{2} \ldots a_{10}}$ that satisfies the conditions of the problem. Clearly, $a_{1}=9$, since $a_{1} \neq 0$. Let $7=a_{k}$ for some $k$. We will consider several cases.

- If $k>6$, then $N=9024687531>A$.
- If $kA$.
- If $k=6$, then $N=\overline{902467 a_{7} a_{8} a_{9} a_{10}}$. If $a_{10} \neq 8$, then $N=\overline{902467 \ldots 8 \ldots a_{10}}>A$. If $a_{10}=8$, then $N=9024675318=A$.

Thus, we have considered all possible cases and confirmed that $A$ is the smallest number.

## Criteria

The largest suitable criterion is used:

15 points. The correct answer is provided with justification of minimality.

8 points. The correct answer is provided without justification of minimality.