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## #1 2012-12-17 05:42:40

jackme12
Member
Registered: 2012-12-17
Posts: 6

### Sum of 3 digits and its compliment to 1000

Can you notice any patterns in the sums of the digits in three digits numbers and the sums of the digits of their complements to 1000.

For example 1000 - 255 = 745

2+5+5 = 12
7+4+5 = 16
12+16 = 28

Bonus points for multiple patterns,
Bonus bonus points for anyone who can find a direct proof.

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## #2 2012-12-17 07:08:19

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

### Re: Sum of 3 digits and its compliment to 1000

For now I only have 10 and 28...

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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## #3 2012-12-17 09:28:22

scientia
Member
Registered: 2009-11-13
Posts: 224

### Re: Sum of 3 digits and its compliment to 1000

Let
be a three-digit number.

Then

.

If

, we write
.

Then the sum of digits is

.

If

, we have
.

If

, the sum of digits is
.

If

, then
and the sum of digits is
.

So the sums are

(i) 28 if

(ii) 19 if
and

(iii) 10 if
and

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## #4 2012-12-17 17:23:49

noelevans
Member
Registered: 2012-07-20
Posts: 236

### Re: Sum of 3 digits and its compliment to 1000

If you cast out nines on the sum of the digits of both numbers together, the cast out digit will be 1.
If c and c' represent the two complementary numbers then casting out nines on these two will
always produce tens complements.  For example COD(255)=3 and COD(745)=7;
COD(111)=3 and COD(889)=7;   COD(100)=1 and COD(900)=9 (9 is equivalent to 0 when casting
out nines.);  COD(257)=5 and COD(743)=5.

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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## #5 2012-12-17 20:39:16

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

### Re: Sum of 3 digits and its compliment to 1000

1000-333=667
3+3+3=9
6+6+7=19
9+19=28
Don't know if correct, jackme12. Reply me if correct or not.

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

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## #6 2012-12-17 22:08:18

jackme12
Member
Registered: 2012-12-17
Posts: 6

### Re: Sum of 3 digits and its compliment to 1000

Wow first time using this forum so I didn't expect an answer as thorough as scientia's.  I wish I had more bonus points to give out.

Julianthemath that is the correct way to start, there are three patterns to be found depending on the initial three digit value you subtract.

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## #7 2012-12-17 22:10:14

scientia
Member
Registered: 2009-11-13
Posts: 224

### Re: Sum of 3 digits and its compliment to 1000

In general, if you have an n-digit number

and

is the n- or (n−1)-digit number that when added to
makes
, then:

If

is the smallest integer such that
,
, then
,
, and
,
.

Thus the sum of digits is

.

Example:

This is equal to

(8 is the no. of digits of N and 4 is the lowest-placed nonzero digit of N).

Last edited by scientia (2012-12-17 22:10:50)

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## #8 2012-12-18 03:57:01

noelevans
Member
Registered: 2012-07-20
Posts: 236

### Re: Sum of 3 digits and its compliment to 1000

And if casting out nines on the sum of the two complementary numbers, the result will still always
be one.

Does anyone use casting out nines anymore?

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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