When last we discussed divisibility rules, particularly, rules for just adding up the digits in a number to tell what it might divide by, we had worked out rules for testing divisibility by eight. In that, we take the sum of four times the hundreds digit, plus two times the tens digit, plus the units digit, and if that sum is divisible by eight, then so was the original number. This hasn’t got the slick, smooth memorability of the rules for three and nine — just add all the numbers up — or the simplicity of checking for divisibility by ten, five, or two — just look at the last digit — but it’s not a complicated rule either.
Still, we came at it through an experimental method, fiddling around with possible rules until we found one which seemed to work. It seemed to work, and since we found out there are only a thousand possible cases to consider we can check that it works in every one of those cases. That’s tiresome to do, but functions, and it’s a legitimate way of forming mathematical rules. Quite a number of proofs amount to dividing a problem into several different cases and show that whatever we mean to prove is so in each ase.
Let’s see what we can do to tidy up the proof, though, and see if we can make it work without having to test out so many cases. We can, or I’d have been foolish to start this essay rather than another; along the way, though, we can remove the traces that show the experimenting that lead to the technique. We can put forth the cleaned-up reasoning and look all the more clever because it isn’t so obvious how we got there. This is another common property of proofs; the most attractive or elegant method of presenting them can leave the reader wondering how it was ever imagined.
Continue reading “Eliminating Your Footprints”
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.
Continue reading “What Makes Eight Different From Nine?”
MJ Howard last week answered my little demonstration that it was easy to tell multiples of two, five, and ten by looking at just the last digit of a whole number, but that there weren’t any ways to tell from just the last digit whether it was divisible by four. He pointed out we could look at the last two digits, and if those were divisible by four, then the entire number would be. This is perfectly true, and it’s only by asserting that I was looking for a rule based on the last digit alone that my forecast of doom about an instant check for divisibility-by-four could be sustained.
Remember the reasoning by which we wrote out a whole number as some string of digits which I call R followed by whatever goes in the units column, which I call a. (I had been thinking of R as in the “rest” of the number, but it struck me over the week that R is also the symbol used in organic chemistry to denote a chain of carbon atoms when one doesn’t really care how many of them are lined up. This interests me as I got on this thread with a set of numbers I called “alcoholic” due to their structural resemblance to organic chemistry’s idea of alcohols.) Since we’re writing in base ten, then, the number written as Ra is ten times R plus a. Ten times R can’t help being divisible by ten, or by any of the factors of ten, which are two and five (and one, which nobody cares about).
Continue reading “How To Recognize Multiples Of 100 From Not So Far Away”