The remainder is whatever is left after the rest has been used up or taken away.
In math, if you know the remainder is the number that's left after you divide, that will help you remember the meaning of this word: a remainder is what you have when others things are gone. If you ate all but one piece of a pizza, the one slice left is the remainder. If you spend all your cash except for $4, that's your remainder. Since remaining means to stay somewhere, a remainder is something that stays put.