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**math9maniac****Member**- From: Tema
- Registered: 2015-03-30
- Posts: 414

Pick any 4-digit number at all (should not have the same digit throughout). Rearrange the digits to form as many 4-digit numbers as possible. From the range of numbers (including the one you first picked), subtract the least from the greatest. If the difference is a 2-digit number add the digits together.

The idea behind this is no matter what your answer will always be 9. Yes try it now.

Only a friend tells you your face is dirty.

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,442

hi math9maniac

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob Bundy

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**Olinguito****Member**- Registered: 2014-08-12
- Posts: 649

The most general theorem of all:

(NB: If *N* contains one or more 0, we allow *N*ʹ to start with a 0. For example, if *N* = 1024, *N*ʹ can be 0241; in this case *N* is a 4-digit number whereas *N*ʹ = 241 is a 3-digit one.)

Proof of the most general theorem of all:

We know that a positive integer is congruent modulo 9 to the sum of its digits. For example, 12345 ≡ 1+2+3+4+5 = 15 ≡ 1+5 = 6 (mod 9) (i.e. it leaves a remainder 6 when divided by 9).

This is based on the fact that is divisible by 9 for all non-negative integers- for all

and adding up the *i* gives

So *N* is congruent to the sum of its digits mod 9. Similarly *N*ʹ is congruent to the sum of its digits mod 9. But the sum of the digits of *N* and the sum of the digits of *N*ʹ are the same since the digits of one are those of the other rearranged. This means

and so their difference is divisible by 9. QED.

*Bassaricyon neblina*

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**math9maniac****Member**- From: Tema
- Registered: 2015-03-30
- Posts: 414

Many thanks bob and Olinguito. I seem to understand bob more. Expecting to hear from you again later.

:-) ;-)

Only a friend tells you your face is dirty.

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**mathaholic****Member**- From: Earth
- Registered: 2012-11-29
- Posts: 3,251

math9maniac wrote:

Pick any 4-digit number at all (should not have the same digit throughout). Rearrange the digits to form as many 4-digit numbers as possible. From the range of numbers (including the one you first picked), subtract the least from the greatest. If the difference is a 2-digit number add the digits together.

The idea behind this is no matter what your answer will always be 9. Yes try it now.

Okay.

1234

1234 1243 1324 1342 1423 1432

2134 2143 2314 2341 2413 2431

3124 3142 3214 3241 3412 3421

4123 4132 4213 4231 4312 4321

4321

1234

----

3087

Well, it's 4 digits but I'll go add it... 3+0+8+7 = 18 (1 + 8) = **9**.

Worth that 10 minute typing. BTW, the name of the topic reminds me of a book I borrow in the library at my school.

Mathaholic | 10th most active poster | Maker of the 350,000th post | Person | rrr's classmate

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