Middle European Mathematical Olympiad 2024 Problem T-7

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.