Does this property have a name?

[Christoph]

In the base-10 system, the digits of any multiple of 9 add up to 9 or a multiple of 9.
Example:
81: 8+1=9 | 81/9=9
27: 2+7=9 | 27/9=3
3258: 3+2+5+8=18, 1+8=9 | 3258/9=362

I'm pretty sure that it works in base-n (where n is an integer > 2) and n-1, but I want to see if there's a proof. To look for a proof, I need a name. Do either of these properties have one?

FWIW, I've taken a year of discrete math, so I know a little bit about proofs. I'm hoping that the proof for this doesn't go way over my head.

 

[nihili]

I don't know whether the property has a name, I've never really seen it discussed anywhere. But it looks like you could do a proof by weak induction on either the length of the numeral or how many multiples of base-1 it is. Here's a sample proof. I'm sure it could be made more elegant with some work, but it gets the job done. I've used base=n+1 rather than n to make it a little neater.

Let S be a number divisible by n expressed in base n+1. If S has only one digit, then that digit is n. Hence the digits of S sum to a number divisible by n

Suppose that S is divisible by n and has digits that sum to n. I now show that S+n also has digits that sum to n.

Let x be the rightmost digit of S. If x is 0, the S+n differs from S only by adding n to the rightmost digit. Hence the digits of S+n sum to a multiple of n.

If x is not 0, then S+n may be obtained by decreasing x by 1 and adding 1 to the preceding digit. If the preceding digit is not n, then the process stops. Since digits have been decreased and increased by the same amount, the sum of the digits is the same in S+n as it was in S. If the second digit (from the right) is n, then it is set to 0 and 1 is added to the third digit. The process continues until we reach a digit which is not n. If all remaining digits are n, then we subtract n from each of them and prefix a 1 to the number. Since we subtracted 1 from one digit, added 1 to another, and either subtracted n or 0 from the remaining digits, the sum of the digits will still be a multiple of n.

nihili

 

[Christoph]

I thank you for prooving my thought.
Although it's not quite what I sought,
I'll decipher your proof,
though I'll feel like a goof,
and perhaps then I'll think myself hot.

 

[Krusty]

I thank you for prooving my thought.
Although it's not quite what I sought,
I'll decipher your proof,
though I'll feel like a goof,
and perhaps then I'll think myself hot.

When toking, one can take lots of hits.
Your brain feels as good as it gets.
But when you come up with a gimmick,
and begin spouting limericks,
I'd say it's time to call it quits.

Ok, so it doesn't quite rhyme, but you get the picture. Whats with the speaking through limerickization today?

EDIT: never mind. Just saw your sig.
EDIT2: you misspelled limerick! Better fix it.

EDIT3: Limerick terrorist, eh? Shouldn't you be wearing a green turban in your pic then? Perhaps a red beard?

 

[Christoph]

I see that you've found out my game.
I suppose that it's me who's to blame.
I have stayed up to late,
and I cannot abate,
so please do not send me a flame.

 

[nihili]

With proofs such as this one it pays,
To induce in the weakest of ways.
With this proof as a tractor,
You'll find that the factors,
Of n work in similar ways.

 

[Johnny Knoxville]

You may be able to use induction,
to prove the property that has nothing to do with chemical combustion,
you may also use proof by contradiction,
although not if you're suffering from alcohol addiction,
rhyming words is very lame,
which is why i will now refrain.

 

[Cheesy Peas]

ooooo dear, i thought this was technical discusion, not rhyming class??

or is this some kind of secret garden?(full of grassyness)

 

[[EG]~NaTz~]

wow nihili looks old,
or so im told,
from the pic in his avatar,
he may play guitar,
or dig up some gold.

ok so mine doesnt read correctly and pardon my opinion but that is just what i thought of when i saw that picture and it looks like it came from the beverly hill billys.(nicest way to say it).

i however never have heard of that property, but if u dont know by monday ill ask my math and physics teachers.

 

[Christoph]

...
i however never have heard of that property, but if u dont know by monday ill ask my math and physics teachers.

Good luck. Nobody I've asked has known what it's called.

 

[axhed]

for a dude wearing a hat of purple
you sure do alot of... errrrr...
the rig is so pimp
the rhymes are so limp
not to mention the rhythm lacks.... uhhh...

 

[Obsidian]

DOH!!

Try posting your question Here

This site covers just about every thing you can possibly think of that deals with math.

I tried to look there for something about it, but i noticed an interesting article dealing with some very deep aspects of infinity and well my mind is kinda messed up for a bit

I would have posted myself but figured since you want to know you should be the one to post

 

[Christoph]

You mean like aleph naught? Fun stuff!
Thanks for the link. I feel like a total geek, but I get the feeling I'll be spending many happy hours there.
[wanders off]Mmmmmmmm, Karnaugh Maps.

 

[nihili]

Aleph naught isn't a big deal. Try wapping your head around the idea of strongly inaccessible cardinals. I still don't quite have that one.

nihili

 

[Christoph]

Aleph naught isn't a big deal. Try wapping your head around the idea of strongly inaccessible cardinals. I still don't quite have that one.

nihili

Is discrete math a hobby of yours, or does it have something to do with the philosophy you teach?

 

[nihili]

I specialize in logic. My Ph.D. minor was in math.

 

[Cjwinnit]

Solved it. (sort of...). It's Sort of inductiony but a solid proof could grow from this i hope.

Start at 1 (we are in Base 10). Find A, the numerical value of the sum of the digits
Add 1, A=2. Keep going..
Add 1, A=3
Add 1, A=4
.
.
Add 1 to 8, A=9.
Add1, A=1 (10 => 1+0=1.)
Keep going...
Every 9th number starting at 1 has digits that sum to 9 as A is cyclic
Values of A go like this.
1,2,3,4,5,6,7,8,[9],1,2,3,4,5,6,7,8,[9],1,2,3,4,5,6,7,8,[9],1,2,3,4,5,6,7,8,[9],1,2,3,4,5,6,7,8,[9],... so every 9th numbers starting from 1 ascending the integers will have digits that will sum to 9.
Base n, n will have digits that sum to 1, A will start at 1 and go to n-1 then go to 1 so in base n where n is bigger than 1 a multiple of n-1 will have digits that will sum to n-1.

YAY!!

 

[IFMU]

First off dont shoot me, I far from belong in here but was way beyond bored and figured I might get a brain wrinkle or 2 if I was to read something in here.
I have no clue what nearly anyone in this thread has said, minus one detail. The math you wrote up on the first post.
It severely caught my eye. I was realllly thinking I was the only person that had thought of numbers in that way. I dont quite follow everything that everyones said, but as far back as I can remember I would do small quick math problems in my head. (as in your 6y/o riding in a car for over 24 hours straight you learn to amuse yourself rather quickly, car tags have tons of numbers and its always been entertaining.?)

[Lame example]: Bored senseless, car goes by, tag # is 385 739. I would add and total these numbers, i.e. 8. No clue if this actually goes with what your sayin or not... [/Lame example]

Ive tried to explain this to others and have always gotten odd stares once I was finished trying to explain. Needless to say I just gave up on it eventually.

Hope ya'll dont mind, but Im subscribing to this thread and hope you can figure out what it is your asking. I would be interested due to the crap I do with this and the on the spot math I always do using this basis.

tries to sneek out of thread before I get noticed and booted......

 

[Cjwinnit]

Here is another property my brain thought up.
Make 2 numbers, A and B.
Find the sum of these digits, F(A) and F(B).
Then make a number X that is the sum the digits of [F(A) + F(B)].

Now, with the original 2 numbers, A and B, add them together
call it C. Find the sum of the digits of C, call it Y.


Note, for any example, X=Y.
Anyone know why?

 

[Cjwinnit]

Example: A = 18, B = 9
F(A) = 9, F(B) = 9
X = 9+9 = 18: 1+8 = 9.
C = 27, Y = 2+7 = 9.
X=Y.