P.I. Find the largest natural number from which it is impossible to obtain a number divisible by 11 by erasing digits.

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If in the decimal representation of a number there is a digit 0 or two identical digits, then by erasing the other digits, we can obtain a number that is a multiple of 11. Therefore, the desired number is at most nine digits long, and all its digits are distinct. The largest such number is 987654321. Let's prove that it satisfies the condition of the problem.

After erasing digits from the number 987654321, we get a number of the form $\bar{a}_{\bar{k}} \ldots . . a_{2}^{-} \bar{a}_{1}^{-}$, where $a_{k}>\ldots>a_{2}>a_{1}$. By subtracting from it a multiple of 11, $\bar{a}_{1}^{-} \bar{a}_{1}^{-}$, and erasing zeros at the end, we get a number of the same form. Continuing this process, we will eventually get a single-digit number, which is not a multiple of 11. Therefore, the original number was not divisible by 11.

## Answer

987654321.