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**farah345****Guest**

Take any 4-digit number, and its reverse, and subtract the two. For example,

7694-4967=2727

Now add all the digits:

2+7+2+7=18

1+8=9

The answer will always be 9. Can someone prove how?

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,357

Hi Farah;

represent the 2 numbers like this.

As you can see the expression on the right hand side is divisible by 9.

Any number that is divisible by nine when you add its digits they will sum to 9.

*Last edited by bobbym (2009-08-11 03:30:47)*

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

Il give you some tips. First prove that the first number (4-digit minus reverse) is divisible by 9. We know (or you can prove that too) that a number is divisible by 9 iff its sum of digits is divisible by 9. So when we first add the digits it must be a number divisible by 9. What possible numbers are there?

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

farah345 wrote:

The answer will always be 9. Can someone prove how?

1111-1111 = 0

"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..."

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**soroban****Member**- Registered: 2007-03-09
- Posts: 452

. .

. .

.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,357

Hi Farah;

It also works on 5 digit numbers:

Again the right hands side is divisible by 9.

*Last edited by bobbym (2009-08-11 10:35:59)*

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,452

Hi soroban,

soroban wrote:

. .. .

.

I tested this in Excel...which found that it is true for 3645 of the 9000 numbers in the range 1000 to 9999.

I didn't try to weed out multiples of 1111 (or anything else).

Found by Excel:**5**-digit results*10890*: 3645 times*10989*: 640 times

Example: 8991 - 1998 = 6993: 6993 + 3996 = 10989

I haven't tried to work out why, but of the numbers that I checked the middle two digits were always 99.

**4**-digit results*9999*: 2880 times*1170, 1251, 1332, 1413, 1494, 1575, 1656, 1737, 1818 & 1998*: a total of 3815 times

**3**-digit results*261, 342, 423, 504, 585, 666, 747 & 828*: a total of 648 times

**2**-digit results*99*: 162

**1**-digit results*Zero*: 90

The digit-sum of the above multiple-digit numbers always reduces to 9: eg,

(a) *10989*'s digit-sum is 27, whose digit-sum is 9

(b) *9999*'s digit sum is 36, whose digit-sum is 9

(c) *747*'s digit sum is 18, whose digit-sum is 9

*Last edited by phrontister (2009-08-11 23:37:37)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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Cool Fact: The lastest kernel is kernel 3.14

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,648

I hear it doesn't make much of a difference.

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no. not much. but it is the pi'th kernel

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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