Writing "leaves remainder r" as an algebraic expression

0:00Find me a number that leaves three when you divide it by five. Eight will do — one five, and three left over. Now find me another one. Thirteen. Then eighteen. Then twenty-three. Notice what you just did. You did not search — after the first one, you added five, and then five again. Every time you add five, the leftover stays exactly where it was. So this question does not have an answer. It has an endless list of them, with a shape.

0:35Before we chase that shape, let us be careful about what a leftover actually is. Divide twenty-three by five and you can write the whole thing as one sentence. Twenty-three is four fives, and three more. Four is how many times five went in. Three is what would not. And that three is not free to be anything. It is never negative, and it never reaches five. Someone could say twenty-three is two fives and thirteen more. The arithmetic is right and the sentence useless, because thirteen still has fives inside it.

1:14A remainder is what is left when you cannot take any more away. That is why it has to be smaller than what you divided by. Back to the list. Three, eight, thirteen, eighteen, twenty-three. Start instead from the multiples of five. Nought, five, ten, fifteen, twenty. Now push every one of them three steps along. That is our list. The family is the multiples of five, shifted. It runs backwards too. Two steps below nought is minus two, which leaves three under five just as three does.

1:54I want to write that entire family in one line. Not one member of it. All of it, at once. Here are six things somebody might write, where the letter k stands for any whole number. Three k plus five. Three k minus five. Three k over five. Five k plus three. Five k minus two. Five k minus three. How do you test one of these honestly? Feed it values of k, and look at what comes out.

2:26Take five k plus three, and start k at nought. Three, eight, thirteen, eighteen, twenty-three. That is our family, and the gap between entries is five, every time. But be careful what counts as passing. It has to produce every member of the family, and nothing else. Now watch three k plus five. Nought gives five. One gives eight. Two gives eleven. Three gives fourteen. Eight is in our family. Five is not, and eleven is not.

3:02The trouble is the three in front of the letter. That number is the step. Every time k goes up by one, the answer goes up by three. And a family that steps by three is not the family that steps by five, whatever you add on the end. So the number multiplying the letter is not decoration. It is what you are dividing by. That kills three k minus five for the same reason, and three k over five, which is not even a whole number most of the time.

3:38Five k minus three steps by five correctly. But run it: two, seven, twelve, seventeen. Right step, wrong family. Those leave two, not three. So the offset must be right as well. Which leaves five k plus three, and five k minus two. Exactly two of the six survive. And the second has a minus in it, which feels like it ought to matter. Run it, starting k at one. Three, eight, thirteen, eighteen, twenty-three.

4:11The same five numbers. Two below a multiple of five, and three above the one before it, are the same place. The offset only matters by where it sits inside one step. That is also why the letter starts at one here, not at nought. At nought it gives minus two, and minus two is not a number of pebbles. There is a picture underneath all this, worth drawing. Lay twenty-three counters in rows of five.

4:43You get four complete rows, and one short row of three. Now do it with any member of the family. You always get some number of complete rows — that number is your k — with a short row of three underneath. Five k plus three, read straight off a drawing. And the short row is always short. It never fills out into a complete one. If it could, you would have taken another row away, and it would not have been left over.

5:13That is the remainder being smaller than the divisor, drawn instead of stated. Now something harder. Find the numbers that leave two under three, and also two under four. The tempting move is to combine the conditions arithmetically — three and four make seven, two and two make four. That is not how it works. Both conditions describe the same number at the same time. So describe one family, then cut it down with the other.

5:46Leaving two under three gives two, five, eight, eleven, fourteen, seventeen. Which of those also leave two under four? Two. And fourteen. Keep going and you get two, fourteen, twenty-six, thirty-eight. A family stepping by twelve. Twelve k plus two. One expression covering all of them. And twelve is not three plus four. It is the smallest number both three and four go into. Here is what that buys you.

6:20I have two numbers. I will not tell you what they are yet. The first leaves three when you divide it by seven. The second leaves five. What does their sum leave under seven? Write them the way we have learned: seven a plus three, and seven b plus five. Add them. Seven a plus seven b, and then three plus five. Everything with a seven in it divides away cleanly.

6:47What is left is three plus five, which is eight — and eight is not allowed to be a remainder for seven. It still has a seven inside it. Take that seven out, and one remains. The sum leaves one. The two numbers were six hundred and sixty-one, and four thousand seven hundred and seventy-nine. Their sum is five thousand four hundred and forty, and it leaves one. We never divided it.

7:17Subtraction works the same way, right up until it does not. Second minus first: five minus three is two, and two is a perfectly good remainder for seven. That difference is four thousand one hundred and eighteen, and it leaves two. No division needed. Now turn it round. First minus second. Three minus five is minus two. And a remainder is never negative. We were careful about that at the start.

7:48So repair it. Add the seven back on. Minus two plus seven is five. That is not a trick to tidy the answer. Adding seven cannot change what a number leaves under seven — it walks you one step along the same family. The rule people half remember is that the remainder of a sum is the sum of the remainders. The half they forget: and then bring it back into range.

8:17Let me finish with a puzzle that is nothing but a list of these conditions in disguise. I have a handful of pebbles. Grouped in threes, one is left over. Paired up, one is left over. Grouped in fives, one is left over. Grouped in sevens, none are left over. And there are no more than a hundred. Read it as four conditions on one number, and sieve. Leaving one under three: that is thirty-four numbers below a hundred. Far too many.

8:50Also leaving one under two: seventeen left. Also leaving one under five: four left. One, thirty-one, sixty-one, ninety-one. And a multiple of seven: ninety-one, which is seven thirteens. One answer. And no searching. One last one, because its shape is worth seeing. A number leaves two when divided by three, three when divided by four, and four when divided by five. Look at those pairs. Two and three. Three and four. Four and five.

9:27Every remainder is exactly one short of what you divided by. So the number is one short of a multiple of three, of four, and of five, all at once. It is one short of a number all three go into, and the smallest of those is sixty. So the answer is fifty-nine. Leaves three under five is not a fact about a number. It names a family — endless, evenly spaced, with the step and the offset in plain view.

9:58Once every number carries its own remainder in the way it is written, the answers fall out of the algebra, and the dividing never happens at all.