Divisibility Rules : In an earlier post on the topic of divisibility rules, I posted some divisibility rules for 2, 4 and 8. I also posted a divisibility rule for 16 which was patterned after the ones for 2, 4 and 8. It turns out that this rule I had “derived” for 16 is completely wrong. And it is in fact, easy to verify that it is wrong, but for some reason, I neglected to verify it (and so far, nobody has pointed out to me that it is wrong using the comments on that post, so it seems as if nobody has bothered to verify it).
Even worse, I implied in that post that the divisibility rules for 2, 4, and 8 can be extended to any power of 2. Given that the extension does not even work for 16, it is obvious that this implication is not correct either.
In any case, revisiting these divisibility rules, and realizing that the rule I published for 16 was wrong, I started thinking about divisibility rules for powers of 2 once again. The simple rule that a number is divisible by 2^n if the last n digits are divisible by 2^n is easy to apply for 2 and 4, and perhaps even for 8. But for numbers like 16 and 32, application of the divisibility rule is almost as laborious as doing the division itself. As such, the main purpose of a divisibility rule, which is to verify whether division without a remainder is possible without actually performing the division, is compromised when you have to check divisibility of the last 5 digits of a number by 32, for instance. And it only gets worse for higher powers of 2.
Based on a lot of thinking and experimentation, I have now come up with what appear to be simpler divisibility rules for powers of 2 that have stood the test of some verification using real numbers (as opposed to verification using the logic that since they are simply an extension of a rule that works for some other numbers, they must be correct!). The rules are explained in this latest post on my blog, and I also show readers how to derive divisibility rules for any arbitrary powers of 2. These derived divisibility rules have the potential to be much simpler to apply than performing full-scale divisions of multi-digit numbers by powers of 2 to verify divisibility.
If you are interested, please visit my blog to read the entire post. This time, I have verified the newly derived rules with a few examples, but this is by no means exhaustive, and it is certainly no proof of the correctness of these rules. If you think you will find these rules useful (at least to stump colleagues or friends at get-togethers), please apply them to some examples and let me know if they work. If you use my derivation to extend these rules to higher powers of 2, let me know whether they work also.
I am not a theoretical mathematician or number theorist, so I don’t know how to mathematically prove that these divisibility rules are correct. If I had been one of them, it probably would not have taken me quite so long to come up with these rules in the first place! But if one of my readers feel like taking on a challenge, please feel free to try to prove the correctness of these divisibility rules. And let me know if you find a proof (of either correctness or incorrectness). Thank you, and good luck!