you divide a magnet, you always get a magnet and another magnet, no matter what.
you divide a knife, you can either get knife/knife or knife/not-knife
you divide a sandwich, you get all three : sandwich/sandwich, sandwich/not-sandwich, or not-sandwich/not-sandwich
if you can name an x where there's no division that results in x/x or x/not-x, you don't have to vote for me for mayor. otherwise you gotta show up and put my name down. that's my campaign