EGMO 2022 Problem 5

For all positive integers $n, k$, let $f(n, 2 k)$ be the number of ways an $n \times 2 k$ board can be fully covered by $n k$ dominoes of size $2 \times 1$. (For example, $f(2,2)=2$ and $f(3,2)=3$.) Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2 k)$ is odd. Proposed by: U.S.A.