Garnierin A.

A natural number $N$ is 999... 99 ( $k$ nines) times greater than the sum of its digits. Specify all possible values of $k$ and for each of them, provide an example of such a number.
Answer: such a number exists for any $k: N_{k}=9 k \cdot\left(10^{k}-1\right)$. Let $9 k=\overline{s_{1}} \overline{-. .}-\bar{t}-\overline{0}-\ldots 0$ ( $s_{t}$ is not equal to 0, there may be no zeros at the end). We will check that the sum of the digits of the number $N_{k}$ is equal to $9 k$. Write the difference between the numbers $9 k \cdot 10^{k}$ and $9 k$, considering that $9 k ] | Difficulty: 4  Classes: $8,9,10$ |
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(in the notation of the left side, there are $k$ nines)