Math - The Joy of 8

Greetings! Long time no write? Oof. Thanks to my class and work, I'd gotten distracted ... a bit. I should remedy that with musings on some lovely properties of a certain number!

For this past Christmas, I got myself and some of my family members a subscription to The Great Courses Plus. I'd used a trial before and found a ton of stuff that I liked - music, math, literature, etc... I went through almost 3 courses on Calculus. Then, I forgot about it for a while. That is, until recently.

One of the courses is called "The Joy of Mathematics," with a various assortment of "Joy of X," with X being some topic in math. One lecture was titled "The Joy of 9." The basic gist of the lecture was exploring the fact that numbers divisible by 9 (i.e. 81, 171, 1611) were all made up of digits that added up to 9. Crazy, right?

Curious about this, I spent some time making a table of numbers to see what the results for the others were. Actually, I did a dumb my first attempt. I thought "Why not include one at the top?" at the same time that I had the numbers going 1-9 on the side. In the end, my chart was just an overly-complicated way of arriving at a basic times table. Derp.

My big OOF reconstructed

However, I eventually figured out a layout to see the lists of each number, with the corresponding "digital roots" next to each one. I found several interesting patterns, actually.

Tables of Numbers with Patterns - weeee

I found that half of the numbers (2, 4, 5, 7, 1) went through all 9 digits once before repeating that cycle. 3 and 6 were a little more interesting - they repeated "3, 6, 9" and "6, 3, 9" respectively. 1 didn't stray from what you'd expect - even after 9, since 10 gave 1 and the cycle began again. And, of course, 9 gave all 9s.

However, one of them stood out to me - 8. Unlike the other cycles, the digits were in order, but unlike 1, they were descending rather than ascending. It was then I realized another interesting property of counting up numbers - the last digit cycled consistently. On its own, that is a "well, duh" type fact. Of course they're going to repeat eventually, I'm adding the same number every time. But I didn't stop at that.

At the beginning, both the last digit and digital root were the same. Writing out the last digits for the rest, I wondered - when do the last digit and digital root coincide?

After the first time, I found that they coincided at 12*8, 23*8, and 34*8. Also, the order they came in was the same as the pattern for 8's digital roots - 8, 6, 4, 2! (Not factorial, I'm just expressing surprise.) It didn't include the zero of the last digit sequence for obvious reasons. Once I got to 45*8, the pattern started breaking down. The last digit was in fact 0, but the number itself was 360 - nonzero - and thus could not provide the matching digital root.

And that's cool, at least to me. No, it's not a neat, easy fact like "9 is divisible by numbers whose digits equal 9" that can be turned into a neat trick for parties, but it's a cool pattern nonetheless. With a little more time, I could find more points of coincidence between 8's digital roots and its last digits, maybe even try to find another hidden pattern, However, that's all from me for now.

8.