KöMaL 2026 A924

For which pairs $(k,l)$ of positive integers is the following statement true: for every positive integer $n$ there exists an integer $i$ with $0\le i\le l-1$ such that $\binom{n}{i}+{(-1)}^l\binom{n}{i+l}+{(-1)}^{2l}\binom{n}{i+2l}+\dots$ is not divisible by $k$?