Math Is Fun Forum

farah345

Guest

nice fact

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

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: nice fact

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)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: nice fact

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?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: nice fact

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

1111-1111 = 0

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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..."

soroban

Member

Registered: 2007-03-09

Posts: 452

Re: nice fact

. .

. .

.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: nice fact

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)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: nice fact

Hi soroban,

. .

. .

.

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)

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"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

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

Website

Re: nice fact

Cool Fact: The lastest kernel is kernel 3.14

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'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.

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: nice fact

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

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

Website

Re: nice fact

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

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'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.