When last speaking about divisibility rules, we had finally worked out why it is that adding up the digits in a number will tell you whether the number is divisible by nine, or by three. We take the digits in the number, and add them up. If that sum is itself divisible by nine or three, so is the original number.
It’s a great trick. We have to want to do more. In one direction this is easy to expand. Last time we showed it explicitly by working on three-digit numbers; but we could show that adding a forth digit doesn’t change the reasoning which makes it work. Nor does adding a fifth, nor a sixth. We can carry on until we lose interest in showing longer numbers still work. However long the number is we can just add up its digits and the same divisibile-by-three or divisible-by-nine trick works.
Of course that isn’t enough. We want to check divisibility of more numbers. The obvious thing, at least the thing obvious to me in elementary school when I checked this, was to try other numbers. For example, how about divisibility by eight? And we test quickly … well, 14, one plus four is 5, that doesn’t divide by eight, and neither does fourteen. OK so far. 15 gives us similarly optimistic results. For 16, one plus six is 7, which doesn’t divide by eight, but 16 does, so, ah, obviously there’s something more we have to look at here. Maybe we need to patch up the rule, and look at the sum of the digits plus one and whether that divides eight.
This may sound a little fishy, but it’s at least a normal part of discovering mathematics, at least in my experience: notice a pattern, and try out little cases, and see if that suggests some overall rule. Sometimes it does; sometimes we find exceptions right away; sometimes a rule looks initially like it’s there and we learn something interesting by finding how it doesn’t.
Obviously, though, “add up the digits of the number, plus one, and check if that divides eight” doesn’t work. It fails utterly for one-digit numbers. It also fails for 24: two plus four plus one is seven and we’ve got another multiple of eight which doesn’t fit the rule. We’re still one short; we’d have to add two to patch it up.
However … for a one-digit number, we shouldn’t have added anything to the sum of the digits. 8 should be divisible by 8 and that’s that. For a two-digit number … well, the numbers in the tens, we had to add together the digits, plus one. 16 implies one plus six plus a surplus one. For the twenties … 24 implies two plus four plus two, which adds up to eight. We might have something after all, especially once we check there aren’t any false positives, that all the other twenty-somethings have digits plus two which don’t add up to a multiple of eight.
Hypothesis: for the thirties, we’ll add the digits and then three. And 32 gives us three plus two plus three again, which is eight, as it should be. The other thirty-somethings add up to numbers like six and seven and ten, none of those divisible by eight, and matching numbers not divisible by eight. For the forties, let’s add the digits and then four. 40 and 48 check out, giving us sums of eight and sixteen respectively; the other forty-somethings give sums that are other non-multiples of eight.
What about three-digit numbers? 100 breaks down to 1 plus 0 plus 0, plus another 1 plus 0, summing up to two. And 100 doesn’t divide by eight. We seem to be on to something. 104, that divides by eight, and sure enough … 1 plus 0 plus 4 plus 1 plus 0 is rats. If only we had two more. 112 yields the similarly disappointing result of adding up to six; again, we need two more. For that matter, 200 has digits which add up, with our duplication, to just four; we need four more there.
For the multiples of eight in the one-hundreds, we end up, with this doubling rules, two short of a multiple of eight. For the multiples of eight in the two-hundreds we end up four short of a multiple of eight. For the multiples of eight in the three-hundreds we end up … well, how about that, six short.
We seem to be developing a rule that says we take the sum of the ones column and twice the tens column and four times the hundreds column, and a little experimentation will probably be convincing that this does work for three-digit numbers. This hasn’t got the elegant simplicity of the threes and nines rule, but on the other hand, it’s not too hard. Of course, it suggests horrible things if we have to work with a four-digit number.
Or does it? The pattern we’ve empirically discovered suggests we would take eight times the number in the thousands column. But eight times anything is a multiple of eight. If we’re looking for how much more this weighted sum of digits is than some multiple of eight, then the thousands-digit-times-eight isn’t going to matter. It’ll increase the weighted sum of digits by some multiple of eight, but not change whether it is a multiple of eight.
The ten-thousands column digit would be multiplied by sixteen … which is to say, by a multiple of eight, so, it doesn’t matter. And the hundred-thousands, the millions, the ten-millions, and so on, all get multiplied by multiples of eight so they don’t matter.
This little experimentation suggests a rule for multiples of eight: take four times the hundreds column, plus two times the tens column, plus the ones column, and add that together. The rest of the digits don’t make a difference. If this sum is divisible by eight, so was the original number. If it isn’t, neither was the original number.
It’s a nice hypothetical rule. We can test it out for the numbers from one to a thousand and see it always holds true there. With a tiny bit of reasoning we can say that that’s all the whole numbers we have to test, and so we can say this rule really and truly works, and that we found it through experimentation, hypothesizing, and adding in that dash of reasoning which lets us claim (correctly) that only the thousand particular cases we might test out need to be tested. And each of those cases proves itself out — if you’ve a lot of time, you might do this by hand, although if you have that much time you’ll probably just have a computer program do it — so the rule checks out. We’ve developed a counting-the-digits rule for testing divisibility by eight.
Still, one might feel this is a little ad hoc, or that the proof is scattered and ugly. So it is. Newly developed mathematical points often are. Perhaps we can go back and re-think things, and find a way to prove it that just works better. It might also give us an idea of what “working better” means.