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#1 2012-12-18 04:42:40
- jackme12
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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.
#2 2012-12-18 06:08:19
- anonimnystefy
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Re: Sum of 3 digits and its compliment to 1000
For now I only have 10 and 28...
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#3 2012-12-18 08:28:22
- scientia
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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
#4 2012-12-18 16:23:49
- noelevans
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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.
#5 2012-12-18 19:39:16
- julianthemath
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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.
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#6 2012-12-18 21:08:18
- jackme12
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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.
#7 2012-12-18 21:10:14
- scientia
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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 (2012-12-18 21:10:50)
#8 2012-12-19 02:57:01
- noelevans
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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.