Divisibility is an odd thing to play with in puzzles. You can can always choose a set of numbers that makes things work nicely, and allows you to create a puzzle with clues you want. It can be more interesting to find a way to work with the 1 through n that you're given. In this puzzle we use the numbers 1 through 6, but include direction to give away which number divides which.
We also introduce clues for when neither number divides the other. These are important to keep track of as without them the solutions would not be unique. I'm including one completed puzzle to show everyone quickly how the clues work, and then an easy, medium, and "frustrating" one. Below are links to the solution, a higher quality PDF, and individual puzzles for those who prefer working with larger cells.