Math Is Fun Forum

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

I just thought of something

Let k be a positive integer.

Then a positive integer N is divisible by k if and only if the sum of the digits of N in base k+1 is divisible by k.

Last edited by JaneFairfax (2008-07-31 02:38:39)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: I just thought of something

Nice fact! (or theorem or postulate or whatever it is)

It's a fun one to prove.

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Why did the vector cross the road?
It wanted to be normal.

ZHero

Real Member

Registered: 2008-06-08

Posts: 1,889

Re: I just thought of something

Can we Hope to read a book "Jane's Discoveries in Mathematics" someday??

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If two or more thoughts intersect, there has to be a point!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: I just thought of something

This one is actually intuitively straightforward. It needs barely anything more than this result: For any positive integer a and any non-negative integer m,

is divisible by

.

Proof: If m = 0,

= 0, which is divisible by

. If m > 0, then

.

So if

in base k+1 – i.e.

– let

.

Then

and

is divisible by

.

So

is always divisible by k; ∴ k divides N

k divides S.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: I just thought of something

Perhaps a more elegant proof would be to interpret a^m - 1 as a polynomial in the variable a.  Then a-1 divides a^m-1 since they both have 1 as a root.  So since:

Where p(a) has integer coefficients, certainly it holds for any integer as well.  I rather like the idea of a field theory argument about integers.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

infinitebrain

Member

Registered: 2008-07-01

Posts: 153

Re: I just thought of something

that's so cool how yall know so much about  Theroies

Last edited by infinitebrain (2008-08-05 05:30:31)

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The best thing about life is you don't know what to expect

Math Genius MT

Banned

Registered: 2008-08-06

Posts: 0

Re: I just thought of something

How did you create that flashing  Space invader (I think..) JaneFairfax

from The Math Genius

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: I just thought of something

Jane has used the [img]tags to put an image into her post.
To insert images, you say "[img](URL of picture here)[/img]".

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: I just thought of something

Space invader?! Where?

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: I just thought of something

I'm guessing they must mean this:

I'm not sure where the problem lies, but occasionally the forum displays pictures far larger than they're meant to be for me (that picture looked huge to me earlier today), so maybe that's what made Math Genius think it was a space invader, rather than just a happy thing.

Edit: Just noticed that Math Genius is ARB, so it's equally likely that he was just talking nonsense.

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Why did the vector cross the road?
It wanted to be normal.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: I just thought of something

When I opened this topic, I also got a
flash of something big on the screen for
an instant.   I think it had a block pattern.
But it only happened the first time I opened
the thread and didn't happen on the 2nd
time I tried.  I'm not joking.

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igloo myrtilles fourmis

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: I just thought of something

Jane's space invader strikes again!
The forum sometimes has trouble displaying [img]'d pictures.

Sometimes it'll display them fine, sometimes it'll display them really huge, and sometimes it'll display them really huge for a moment and then correct itself. That last one probably happened to you.

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Why did the vector cross the road?
It wanted to be normal.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: I just thought of something

Yes, you are correct.  It is the GIF files magnified indeed.
It doesn't happen in my Safari browser, but it does happen
in my old IE5 browser at times.  I have now seen it with
the colors inside the black block outlines in 2 different threads.

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igloo myrtilles fourmis

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: I just thought of something

more generally, let d and k be positive integers such that d|k, then N is divisible by d iff the sum of the digits in base k+1 is divisible by d.