Question 91
Define glueing of positive integers as writing their base ten representations one after another and interpreting the result as the base ten representation of a single positive integer.
Find all positive integers $k$ for which there exists an integer $N_{k}$ with the following property: for all $n \geq N_{k}$, we can glue the numbers $1,2, \ldots, n$ in some order so that the result is a number divisible by $k$.
Remark. The base ten representation of a positive integer never starts with zero.
Example. Glueing 15, 14, 7 in this order makes 15147.
Find all positive integers $k$ for which there exists an integer $N_{k}$ with the following property: for all $n \geq N_{k}$, we can glue the numbers $1,2, \ldots, n$ in some order so that the result is a number divisible by $k$.
Remark. The base ten representation of a positive integer never starts with zero.
Example. Glueing 15, 14, 7 in this order makes 15147.