The subtraction game

In the subtraction game a single pile holds some tokens, and each turn a player removes between $1$ and $k$ of them. Whoever takes the last token wins. The losing positions are the multiples of $k+1$: if the pile is a multiple of $k+1$ on your turn, any move lets the opponent restore a multiple of $k+1$, and you are eventually handed the loss.

So the winning move is always to leave a multiple of $k+1$. Only the pile size modulo $k+1$ matters, not its full size, which is modular arithmetic doing the work.