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.