1. Find the smallest natural number $n$ such that the sum $n+2n+3n+\cdots+9n$ equals a number whose digits are all the same in decimal notation.
I/1. Let $s=n+2n+3n+\cdots+9n=45n$. Since 5 divides the number $s$ and since $s$ has all digits the same in decimal notation, all its digits are 5. Since 9 also divides the number $s$, $s=555555555$ and therefore $n=12345679$.

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