Is 2,904,897 A Prime Number?

This little trick may win a bar bet someday and it's inexplicable on the surface, so is cool just for that. Well, it's not EVEN, so isn't a multiple of 2, doesn't end in 5, so isn't divisible by 5, but that leaves a lot of ground.
In case you forgot, you only have to divide a prime "candidate" by every interger up to the square root of the number; if you've not found a factor by then there won't BE any. But that's a lot of work if you don't get lucky. Here's the "trick": Add the digits of the candidate [2+9+0+4+8+9+7 = 39]. If the sum is divisible by 3, then so is the candidate.Sure, if you ACTUALLY divided by 3 you'd discover it's not prime, but what if the candidate isn't so small. It's a lot of dividing. Keep adding the "sum" numbers when the numbers are big, like 39932133621 as a sum for a large candidate, paring it down to size [3+9+9+3+2+1+3+3+6+2+1 = 42] then [4+2 = 6] until it's managable.
freeed
66-70, M
10 Responses May 26, 2010

any book references to this trick?

I was always fascinated at Beiler's book on Theory of Numbers which is one of the more unusual ones by Dover. Lost a little charm after going back to it after many years, but it still had a very interesting organization, and 5% of the time you wished one slim chapter said a lot lot more...

Incidentally, the other factor, 968299, is prime

It IS!

@DV866: That is actually a more general effect, and a quite practical one to use in everyday life.<br />
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Sum the digits (casting out nines) in numbers being added, subtracted, multiplied, or divided in any combination no matter how complex, then perform the same sequence of operations on the single digits. You can do it quickly in your head as you run your eyes over the numbers, you never need to remember more than a single digit! The single digit answer you get will match the sum of the digits in the full answer every time -- if it doesn't the full answer is incorrect!<br />
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This is a great way to increase your score on standardized test like the ACT, SAT or GRE. Set up the problem, get the single digit answer, then quickly select whichever A, B, C, or D answer has the correct single digit result without wasting several minutes doing the full calculation like everyone else.<br />
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It's also useful at the checkout counter, balancing your checkbook, doing your taxes, etc..<br />
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http://en.wikipedia.org/wiki/Casting_out_nines

Dang! That's news to me and music to my ears. Um, in what field might those 11's be?<br />
Now all we have to do is come up with a way to do something like that for prime numbers.

something to do with binary ;)

There's one for 11 too, although it's a bit more complicated than the others. You first add up every alternate digit, then add up the rest of the digits separately and subtract the two results. If the answer is a multiple of 11, it's divisible.<br />
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Eg:<br />
147763<br />
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add up the alternate digits, and you get 1+7+6 = 14. Then add up the rest of the digits separately and you get 4+7+3 = 14. Subtract the two and you get 14 - 14 = 0, which has 11 as a factor! <br />
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Similarly, if you have 135802469, you'd do<br />
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1+5+0+4+9 = 19; and<br />
3+8+2+6 = 19<br />
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19-19 =0, which means the number is divisible. <br />
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It's pretty useful in my field, since we deal with multiples of 11 a lot.

That's facsinating--I never knew that before. Thank you!

Out of curiosity what field do you study?

2,904,897x9 = 26,144,073 => 2+6+1+4+4+0+7+3=27 => 2+7=9!!!!! That IS amazing!<br />
My spiritual teacher has said "numbers and mathematics are the vibrations of The Self [God}". See how numbers vibrate???<br />
Thanks DV866...

One I like is what I call the rule of 9's, any number mulitplied by nine ,the result will always break down to nine, E.G. 9 x 3= 27 - 2+7 =9, 9 x 5= 45 - 4+5=9, get the idea, it doesn't matter how big u go it always comes back to 9. 9 x 376=3384 , 3+3+8+4=18, 1+8= 9 Amazing, well I think it is anyway.

FFW: none of which I am aware. To state the obvious, any number ending in 5 is divisible by 5. In high school I generated a table of powers of 5 and discovered some VERY interesting relationships between the digits listed in the table involving group theory.

I had guessed it! I learned that adding trick in the 5th grade But i didn't know we could do what you've done with 4+2=6<br />
:)<br />
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Btw, is there a trick for knowing numbers divisible by 7?