Even sum of digits!!!

anonimnystefy · 2012-02-06 02:45:23

hi guys

is there a formula for the number of numbers between a and b such that the sum of digits of each number is even?

Alex23 · 2012-02-06 05:15:49

What kind of numbers can a and b be?

anonimnystefy · 2012-02-06 05:47:49

they're natural numbers between 1 and 2^30 and a<b.

Alex23 · 2012-02-06 06:02:42

One last question. You want for a single sum, for example 317 --> 3+1+7 = 11, or until there is a single digit, 11 --> 1+1 = 2 ?

anonimnystefy · 2012-02-06 06:11:37

just the single sum. 317-->11

bobbym · 2012-02-06 07:18:25

Hi anonimnystefy;

Yes, there are generating functions for that. Please show an example so that I can clearly understand what you want.

anonimnystefy · 2012-02-06 07:39:35

for 6 and 12 we have:

6,8,11 so the number would be 3.

bobbym · 2012-02-06 07:46:42

You want [1, 2^30) or [1,2^30]?

Let's worry about that later. This is one idea to solve the problem, there are others.

We start with the generating functions:

For computation we will use the expanded forms because there is a nice theoretical result using them.

What does f(x) represent. It represents all the two digit numbers! The following technique uses the roots of unity to only extract the even numbers.

So there are 45 two digits numbers. We continue the process for all 3 digit numbers.

So there are 450 three digit numbers.

One more and we will be able to conjecture a pattern.

There are 4500 four digit numbers.

So are sequence looks like this 4, 45, 450 ,4500 ...

Can you take it from here?

anonimnystefy · 2012-02-06 08:43:37

i don't think so.i want the numbers with even sum of digits between a and b not between 1 and 2^30.

bobbym · 2012-02-06 08:48:25

What is a and b? Variables? Cause we will use the same technique.

Where did he go? I hope he was not abducted by aliens?!

Are you saying you want the number of solutions to this diophantine equation?

With the constraints that a < b and 1 ≤a,b ≤ 2^30 and n = 0,2,4,6,8,10,12...?

We will use the techniques of experimental math:

Using this formula and taking a = 1 and b = 2^30 the number of soultions to that is:

288230375614840832

If you want say a = 10 to b = 2^30 do you see how to get that answer?

Alex23 · 2012-02-06 09:24:30

As you count they generally alternate odd, even, odd, even etc. But for every 10 numbers (starting with 1) they repeat. E.g. 9 (odd) , 1+0 (odd), and 3+9 (even) , 4+0 (even)

Generally it looks like for every 10^(n) for odd n there is a repeat. E.g. 9+9+9 (odd) 1+0+0+0 (odd). But 9+9 (even) and 1+0+0 (odd).

It remains for it to be proven, then apply iteration and modular arithmetic and get the algorithm.

anonimnystefy · 2012-02-06 09:44:51

hi bobbym

no still not what i want

hi Alex23

that's what i thought of too but the code would then look...i'm sure there is a better way.

bobbym · 2012-02-06 09:52:49

no still not what i want

Please clarify.

gAr · 2012-02-06 17:46:28

Hi anonimnystefy,

As an example, are these the numbers you want in [6,188] ?

6, 8, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 110, 112, 114, 116, 118, 121, 123, 125, 127, 129, 130, 132, 134, 136, 138, 141, 143, 145, 147, 149, 150, 152, 154, 156, 158, 161, 163, 165, 167, 169, 170, 172, 174, 176, 178, 181, 183, 185, 187

total: 91

bobbym · 2012-02-06 19:20:47

Hi gAr;

That is what I am getting now but still no formula for a and b.

gAr · 2012-02-06 19:32:39

Hi bobbym,

Okay, I was waiting to get it clarified.
Your g.f is good!
But I too don't have a formula yet for any [a, b]

bobbym · 2012-02-06 19:51:39

Hi gAr;

I agree about getting it clarified. He will be up in a few hours and then we will know.

anonimnystefy · 2012-02-07 08:57:59

hi gAr

those would be the numbers.

to both of you,

i noticed that that number is always either (b-a) div 2,(b-a) div 2 +1 or (b-a) div 2 -1.

bobbym · 2012-02-07 09:10:38

Hi anonimnystefy;

If that is correct it is better than what I was working on.

anonimnystefy · 2012-02-07 09:13:03

i don't understand your sentence.

bobbym · 2012-02-07 09:15:10

Hi anonimnystefy;

Poor typing. Left out a than.

anonimnystefy · 2012-02-07 09:20:56

well i'm sure it's true.you can notice in every ten numbers(1-10,11-20,21-30,...) exactly five numbers have an even sum of digits.i'm just not sure if it can be (a-b) div 2 -1.

bobbym · 2012-02-07 09:24:25

Hi;

(1-10) has 4.

Alex23 · 2012-02-07 09:24:56

anonimnystefy wrote:

hi gAr
those would be the numbers.
to both of you,
i noticed that that number is always either (b-a) div 2,(b-a) div 2 +1 or (b-a) div 2 -1.

That seems to be true. I didn't find any counter example yet after 10. Those repeats offset each other. If I didn't have exam tomorrow I would work with it. Modular arithmetic seems to do the trick and hopefully we can prove also why the formula works to be sure.

Last edited by Alex23 (2012-02-07 09:27:15)

anonimnystefy · 2012-02-07 09:26:38

hi bobbym

yes,you are right.but still...i think it's always true.

Math Is Fun Forum

#1 2012-02-06 02:45:23

Even sum of digits!!!

#2 2012-02-06 05:15:49

Re: Even sum of digits!!!

#3 2012-02-06 05:47:49

Re: Even sum of digits!!!

#4 2012-02-06 06:02:42

Re: Even sum of digits!!!

#5 2012-02-06 06:11:37

Re: Even sum of digits!!!

#6 2012-02-06 07:18:25

Re: Even sum of digits!!!

#7 2012-02-06 07:39:35

Re: Even sum of digits!!!

#8 2012-02-06 07:46:42

Re: Even sum of digits!!!

#9 2012-02-06 08:43:37

Re: Even sum of digits!!!

#10 2012-02-06 08:48:25

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#11 2012-02-06 09:24:30

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#12 2012-02-06 09:44:51

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#13 2012-02-06 09:52:49

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#14 2012-02-06 17:46:28

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#15 2012-02-06 19:20:47

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#16 2012-02-06 19:32:39

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#17 2012-02-06 19:51:39

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#18 2012-02-07 08:57:59

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#19 2012-02-07 09:10:38

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#20 2012-02-07 09:13:03

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#21 2012-02-07 09:15:10

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#22 2012-02-07 09:20:56

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#23 2012-02-07 09:24:25

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#24 2012-02-07 09:24:56

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#25 2012-02-07 09:26:38

Re: Even sum of digits!!!

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