I have now heard from two real math people (that is, people who do it vocationally rather than just recreationally) who confirm that the eleven trick does work as advertised. The first, Matty boy, left me a comment that said it would work in any base, just as the nines trick does. Ben, a doctoral student that also spends a lot of time in my favorite coffee shop, worked through a couple of proofs with me to show me why it would work in any base B. I wish I could replicate them here- and if I had a scanner to bring in math scribbles as images, I would. But I don't, so I can't, so I won't. I envision trying to type in a bunch of math notation and shudder. (My keyboarding skills are best described as "in need of remediation.")
I do want to point out that these "stupid math tricks" that I will spend occasional time discussing are neither "stupid" nor simply "tricks." I chose the word trick because, really, who doesn't enjoy knowing a few "magic" tricks? I don't consider these procedures just "tricks" because they can, on occasion, be very useful. In the context of one of my undergrad lab jobs, people got into the habit of calling out arithmetic problems to me. ("Hey, if there's .62 grams of carbon in a 5.4 gram sample, what's the percent C?" "Well, umm, about 11 and a half.") I'm convinced this got me a couple of raises. I could generally answer within a few seconds then, though due to lack of practice and an aging mind, I'm slower now. It's partly the lack of practice I want to compensate for now, but mostly I want to pass on the tricks. They're simple and easy to pull off inside your head, without looking for the nearest calculator, and they can truly be useful. I want to encourge people to be facile with math and arithmetic.
I chose the word "stupid" first to make the topic seem a little less threating (these tricks are easy to use), and second as an allusion and homage to Letterman's stupid pet tricks- which were often quite amazing. Just as some of the stupid math tricks are.
Since I brought up the topic again, but don't want to spend a whole lot of time on it, I will simply review some divisibility tests that you should know. (Here, the word "number" refers to a whole number or integer. I'm not dealing with fractions in this post.)
1- Every number is divisible by one: no test necessary.
2- Even numbers (those ending in either 0, 2, 4, 6 or 8) are divisible by 2.
3- Sum the digits of the number; if the sum is divisible by 3, so was the original number.
4- Look only at the last two digits (tens and ones). If that number is divisible by 4, the whole number is. If you don't know immediately, subtract a multiple of 20 (20, 40, 60, 80) to get a number between 0 and 19. (for example, 76, use 60 to get 16, or for 52, use 40 to get 12). If the result is divisible by 4, the original number is. This works because each place 100's and higher is divisible by four.
5- If the last digit is 5 or 0, the number is divisible by 5.
6- If the number is divisible by both 2 and 3 (use those tests), it's divisible by 6.
7- I've seen some methods for 7, but don't recall them right now; they tend to be involved, and so far I've just found it easier to actually do the division and see.
8- Look only at the last three digits (100's down). If that number is divisible by 8, the whole number is. If you can't tell, look at the 100's and 10's places; treating that as a two digit number, subtract the largest multiple of 4 you can. Stick the ones digit back on. If the result is divisible by 8, so was the original number. For example, test 23697776: we only look at 776. If you can't tell with a glance that 776 is divisible by 8 (it happens to be 24 less than 800), look at the 77 part. Subtract 76 (largest multiple of 4) leaving 1, slap the 6 back on for 16. 16 is divisible by 8, so the original number is too. This works because each place 1000's and higher is divisible by 8.
9- Sum the digits; if the result is a multiple of 9, so was the original number.
10- Gimme a break.
11- See this discussion (and why the 9 trick works) under yesterday's post "Is that Divisible by Eleven?"
12- Run the tests for 3 and 4; if they both check out, the number is divisible by 12.
13- and 14- I don't know. I'm thinking about 7; if I find a good one, combine that test and 2 for a 14 test. I'll look and think about 13, but that's an obnoxious number.
15- Run 3 and 5 tests, if both yes, it's divisible by 15.
16- Your turn: look at the test for 4 and 8 and see if you can figure this out for yourself.
Public service announcement: The author strongly advocates the practice of safe math by responsible adults ages three and up. If you think you might engage in math, always carry backup paper, pencil and an eraser. For people with impaired vision, enjoyment is often enhanced by use of prescription safety goggles.
Field of Dreams, Field of Dirt
2 days ago