7. Find the smallest natural number $N$, such that the decimal representation of the number $N \times 999$ consists entirely of sevens.
Answer 778556334111889667445223

Solution: $N \times 999=77 \ldots 7$, then $N$ is a multiple of 7, denote $n=N / 7$. We get $999 n=$ $1000 n-n=11 \ldots 1$, so $1000 n-111 \ldots 1=n$. Write it as a column subtraction and repeat the found digits of $\mathrm{N}$ with a shift of 3 to the left $* * * * * * * 000$

1111111111

$* * * * * * 889$

$* * * * 889000$

1111111111

$* * * 777889$

777889000

111111111

6667778889

Notice that the digits repeat every 3, so we get

$n=111222333444555666777889$.

Therefore, $\mathrm{N}=7 \mathrm{n}=778556334111889667445223$.

For graders: an alternative approach is to take the number 111... 1 and start dividing it by 999 in a column, until it divides evenly. I think you shouldn't deduct points if at the very end the number n was incorrectly multiplied by 7.

## Variant $2-\mathrm{b}$