Divisibility Rules For Powers Of 2 Redux

Divisibility Rules : So, last week, I thought I had come up with general rules for checking divisibility by powers of 2 that involved splitting the last few digits of the number into chunks of various lengths, multiplying them by various coefficients, adding them up and seeing if the sum is divisible by the power of 2. I also mentioned that I could not verify whether those rules were correct because I was not a theoretical mathematician or number theorist.

Actually, you don’t need to be either of those to verify divisibility rules or derive new ones. All you need is to know the basics of modulo arithmetic. In fact, I had already used modulo arithmetic once before to explain why divisibility rules derived using osculation work. I had forgotten about that when I did my latest work on divisibility rules for powers of 2.

Fortunately, I remembered that work right after publishing my previous post. It was quite an eureka moment when that happened! I spent some time using modulo arithmetic to verify whether the rules I postulated actually work. Some of them do, some of them don’t. And I also used modulo arithmetic to derive rules for divisibility by powers of 2. There are a large number of such rules, and I have now put together tables of such divisibility rules in the latest post on my blog.

The exercise has been a lot of fun for me. The sense of discovery you get when you figure out that you can build on something simple and basic, to explore something seemingly unconnected and much more complex, is priceless. If you have been intrigued by my forays into divisibility rules, I encourage you to visit my blog, and read my latest post. This time, you just have to take them and apply them. You don’t have to try to verify them, find counterexamples or anything like that. Thank you, and good luck!