The alternating-sum test for 11

0:00You have a test for nine. Add up the digits. Now try it on eleven. Take five hundred and seventy-two, which is a multiple of eleven. Its digits add to fourteen, and fourteen is not a multiple of eleven. So adding the digits does not work at all. But do not throw the method away, because the method itself still works. Only one thing has to change. For nine, every place value sat one PAST a multiple of nine. That is why every digit contributed itself and you could just add.

0:39For eleven, the places do not all lean the same way. One sits past. The next sits short. The next sits past again. Every digit still contributes itself. It just contributes with a sign now. So let us go and look at the places. Start with the units place, which is worth one. One is nought elevens, plus one. It runs one past a multiple of eleven. Call that plus one.

1:09Now the tens place, worth ten. Ten is not one past anything useful. It is one SHORT of eleven. Write it that way: ten is one eleven, minus one. Call that minus one. And that is already the whole difference from the nines argument. Two places in, the lean has changed direction. The hundreds place. A hundred. Nine elevens is ninety-nine. And a hundred is ninety-nine plus one. So a hundred runs one past. Plus one again.

1:45Notice what has happened. Past, short, past. The sign has flipped twice. And notice what has not happened. The offset is still one. Not two, not five. Always one. That is what keeps a digit contributing only itself, and it is the same thing that made the nines test possible. A thousand. Ninety-one elevens is one thousand and one, so a thousand is one short. Past, short, past, short. Twenty-four places checked, then twenty-four more, and it never breaks.

2:23Here is why it cannot break. Take a place that runs one past and multiply by ten. One past, ten times over, is ten past. But ten is one short of eleven, so ten past a multiple is one short of the next one. The lean flips. Every single time. That is the alternation, and it is a consequence, not a coincidence. Compare nine, where every place runs one past and nothing ever flips. That is exactly why the nines test needs no signs and this one does.

3:02Now read a number straight off its digits. Four hundred and sixty-two. Four hundreds. Each hundred runs one past, so four hundreds run four past. Four ninety-nines is three hundred and ninety-six, and four hundred is that plus four. Six tens. Each ten falls one short, so six tens fall six short. Six elevens is sixty-six, and sixty is that minus six. Two units. Each unit runs one past, so two units run two past.

3:38Now collect. Four past, six short, two past. Four minus six plus two is nought. So four hundred and sixty-two sits exactly on a multiple of eleven. And nothing there was a division. That gives you a procedure, and it is worth writing down as two columns. One column for the digits whose places run past. The units, the hundreds, the ten-thousands, and so on up. One column for the digits whose places fall short. The tens, the thousands, and so on.

4:17Total each column. Then subtract the shortfall total from the excess total. Three steps, and not one of them is a division. Try it on a six-digit number. Three, two, nought, one, eight, five. The places that run past hold two, one and five. Those add to eight. The places that fall short hold three, nought and eight. Those add to eleven. Step three. Eight minus eleven. Eight minus eleven is minus three.

4:51And a lot of people stop right there and assume they have made a mistake. They have not. Minus three is a perfectly good answer, and it is telling you something precise. It says the number falls three SHORT of a multiple of eleven. That is not the same as a remainder. A remainder counts how far you are PAST the multiple below you. What the difference reports is a signed offset. Which multiple you are nearest, and which side of it you are on.

5:25Being allowed to come out negative is not a flaw in the test. It is the test carrying information a remainder cannot carry. So the number is three short. Fine. We will convert that in a moment. First, the same test written far shorter. Put a plus in front of the units digit, then a minus, then a plus, alternating all the way to the left. Then just evaluate it. No columns, no totals, one line.

5:57On the same six-digit number that gives minus three again. And that is not luck. It is the same procedure. Sorting the digits into two columns and then subtracting IS putting alternating signs on them. Swept over three thousand numbers, and three thousand more up around half a million, the two forms agree every single time. So there are not two tests here. There is one test written two ways.

6:29Now convert. Our number is three short of a multiple of eleven. If you are three short of the multiple above, how far past the multiple below are you? Eleven minus three. Eight. So the remainder is eight. Add eleven to a negative total and you have it. Is that step worth a scene of its own? Yes, because of how often it bites. Of three thousand small numbers, the alternating total is not the remainder for nine hundred and twenty-eight of them. Of three thousand six-digit numbers, it fails for one thousand seven hundred and thirty-eight.

7:10Roughly a third, then well over half. The longer the number, the more likely you need that last step. One more case, and it catches people out. What if the difference itself comes out as eleven? Take ninety thousand nine hundred and four. Its alternating total is twenty-two. Twenty-two is not nought. But twenty-two is a multiple of eleven, and that is enough. The number is a multiple of eleven.

7:42Take eight hundred and fifty-seven thousand and seventy-six. Its total is minus eleven. Also a multiple. Also a yes. Nought, eleven, minus eleven. Across three thousand numbers those are the only multiples of eleven any total ever reaches, and every number reaching one of them is a multiple. So the rule is not that the total must be nought. It is that the total must be a multiple of eleven.

8:15Two ways to get this wrong, both tempting. First, throwing away zeros. A zero adds nothing to its column, so why keep it? Because it still holds a place, and every digit to its left is sitting one place further along than you think. Drop the zeros from ninety thousand nine hundred and four and you get nine hundred and ninety-four, whose total is four, not twenty-two. A multiple has become a non-multiple.

8:46Second, starting the signs at the left-hand digit instead of the units. That agrees perfectly on all nine hundred three-digit numbers, which is exactly why it survives. On four-digit numbers it flips every sign. It still spots multiples of eleven correctly every time, and it reports the offset the wrong way round almost always. The signs are anchored at the units because the units place is the one that runs one past. Everything else follows from there.