I had a math hypothesis which may or may not be too simple for your minds to bother with.
While counting for my children by threes every night (to put them to sleep), I noticed that all numbers in which there are three alike in a row, are always divisible by three.
Thus, 111, 222, 333, 444, 555, 666, 777, 888, and 999, are divisible by three.
Secondly, I wondered if this would be true for all the numbers. Would 3333 and 5555 be divisible by four? This, sadly, does not seem to be the case. So it seems to be true only for 3s, up to 999. Why is this the case?
There may be other patterns for which it works, but I wondered why it works so well for threes, and not for other numbers. My original hypothesis was that any time therefore that you had eight of the same numbers across, you would be able to divide it by 8, and get a clean result.
Thus, 77777777 should be divisible cleanly by 8. However, this does not work out. There might be some other pattern at work in some numbers where this works, but it just doesn't come out neatly for any of the others that I've tried except for 3s. Isn't this sad?
Even if you try to do twos up to 99, it doesn't work, because you hit the odd numbers of 33, 55, and 77.
The world is so sad!
Except for the fact that our snipers cleanly blew the heads off those 3 pirates yesterday, I would be rather depressed. That was a neat trick! We turned them all into zeros, in a twinkling. Which is also sad, if you think about it too long, or think about it in the framework of the brotherhood of man, or also, poor Africa, instead of merely thinking that the brave captain had been saved, and think for once American military strength actually accomplished something, instead of being merely stuck in another debacle, where the two sides blathered on and on, lawyers got involved, and no real justice resulted.
I wish things would always work out so neatly as that anti-piratical exploit -- three shots, three clean hits, and the captain's wife has her hubby back.
Oy vey.
I hope that you all enjoyed Easter. I went four times, and my daughter who's now 9, wanted to go a fifth time. She got her first communion, and now can't get enough of the wafer and the wager, with the wine.
Tuesday, April 14, 2009
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11 comments:
There are some neat results nevertheless that are fun to think about, and which show a weird symmetry. For sixes, for instance, you get odd results like this: 555555 divided by 6, equals
92592.5
Thus, 925 repeats twice, in the answer.
But it's broken up by that dastardly little decimal.
What's that about?
I don't know, but it's fun to think about.
Regarding number theory I have no idea Kirby other than I think that is neat. The most number theory I ever got into was spelling naughty words on my calculator in elementary through highschool. (It took me a while to grow up)
However I am happy about the pirates getting their just rewards even though the question of if violence is justified by Christian standards and morality leaves me wondering sometimes late into the night.
Kirby,
The reason it works for 3 is because, in general, a number is divisible by 3 exactly when the sum of its digits is divisible by 3. And adding the digits in a number like 555 gives you 3*5, which of course is divisible by 3.
A number is also divisible by 9 exactly when the sum of its digits is divisible by 9. This is the basis for "casting out nines". It also means any number made from repeating the same digit 9 times will necessarily be divisible by 9.
This works because of the way the base ten number system works. A number like (abcd) means a*1000 + b*100 + c*10 + d*1. But each power of 10 is one higher than a multiple of 9 (and hence of 3): 1000 - 1 = 9 * 111; 100 - 1 = 9 * 11, etc. So the number (abcd) can be written
999*a + 99*b + 9*c + a + b + c + d.
Since the first three terms are clearly divisible by 9 (or 3), the whole number is divisible by 9 (or 3) exactly when a+b+c+d is.
It doesn't work for other numbers because the numbers 9, 99, 999, 9999, etc. aren't all divisible by any other numbers (except, trivially, 1).
Thanks for this explanation, George. I shall ponder it for a bit. It's amazing to see a mathematician's familiarity with this language of figures, and how easily the figures are combined.
I love it! It's like magic.
P.S. Do you think there's any specific reason I got a repeating number when I divided 555555 by 6, or was that just a fluke? 92592.5 was the result.
I did this on paper. I should have checked it with a pocket calculator but I can't find mine.
Kirby,
P.S. Do you think there's any specific reason I got a repeating number when I divided 555555 by 6, or was that just a fluke? 92592.5 was the result.It's basically because each group of three 5's is exactly divisible by 3, so dividing the first 555 group by 3 yields 185, and then dividing the second 555 group by 3 yields another 185. Dividing each of these by 2 then gives the two groups of 925, since 185/2 is 92.5. You'll get something similar whenever you "repeat" any three-digit number which is either divisible by 6 or is divisible by 3 and is less than 600 (so that the resulting pieces don't interact). For example:
234234234/6 = 39039039,
824824824/6 = 104104104,
465465465/6 = 77577577.5, but
873873873/6 = 145645645.5 (which comes from 145500000+145500+145.5, which gives the "interacting" pieces).
And really there's nothing special about dividing by 6 here, or using groups of length 3. For example, consider:
864586458645/14 = 61756175617.5
You just need the resulting fractions to not "repeat" (like in 1/3 = 0.333333...), and the lengths to be such that the pieces don't "interact" upon division.
I thot things were
"one nation
indivisible
w Liberty and Justice
for All"
next thing y'all will tell us: 3 has a religious significance
and
on CNN a photo (close up) of that little boat clearly at least 7 bullet holes in the shell 4 more than 3
maybe things will be absolutely clear in the movie ?
OH. there already is a movie out:
Dr Danger ..Somalia
it s free... via hulu
speaking of ,,,
Dr. Arnot...Dr. Danger in
Somalia...now. brief 20 seconds commercials not many and need to click "full screen" once n awhile
http://www.hulu.com/watch/12629/dr-danger-somalia
An american/ a foreigner will last 1 or 2 minutes on the streets without
armed guards then be taken hostage.
Ransom is body weight x $1000
If they end up killing you they will blame it
on al qaida and ask for $$$$ help in pursuing them
there is also a movie "out there" Black Hawk Down
this is an anarchy ..not just talk about ["it"]
I like George - I wish he would comment more.
He always knows what he's talking about, and has a steady-but-not-boring style. But maybe that's because he doesn't comment unless he knows what he's talking about.
I like it when George comments, too. I got this in my email from another math professor. I've read it several times, and I think it covers some of the same points that George covered.
"Regarding the math question. Here's a relevant fact : 10 mod 3 = 1, i.e.,
if you divide 10 by 3, you get a remainder of 1. This means that it is very easy to compute remainders of big numbers when divided by 3 by simply adding up their digits. Consider 754. What is it's remainder when divided by 3?
7 + 5 + 4 = 16, 1 + 6 = 7, and 7 when divided by 3 gives a remainder of 1. So the remainder of 754 divided by 3 is also 1.
You considered a special case: numbers of the form nnn. If you add up the digits, you get 3n, which when divided by 3 gives a remainder of 0. So numbers of the form nnn are always divisible by 3... Even if n is more than a simple digit, e.g., 123412341234 (n = 1234) must be divisible by 3.
Now 10 mod 9 is also 1. So the digital sum trick works here too. As before, the digital sum of 754 reduces to 7, so the remainder of 754 when divided by 9 is 7.
There are other well known divisibility rules, which work in similar ways. For example, 2 divides 10. So if you want to know if a number is divisible by 2, you only need consider the units place: 12131341324352452452 is divisible by 2 because 2 (the last digit) is. Likewise 4 divides 100, and 8 divides 1000. Thus, 532 must be divisible by 4, because 32 is.
There are lots of patterns in numbers!"
This was from a top math prof at an Ivy League school who also happens to be a Lutheran, so he lurks at my blog from time to time.
Mathematical minds fascinate me! I'm still struggling with elementary algebra, but I thought I had conceived some kind of giant new number theory. Ha ha! At least I got a response, which was fun to get. I didn't know what the term "mod" meant.
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