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   Author  Topic: Number Theory  (Read 775 times)
anonymous
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Number Theory  
« on: Jul 7th, 2003, 12:10am »
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Find all positive integers n such that 0,1,...,n-1 can be rearranged into a(0),a(1),...,a(n-1) that satisfied the following condition: the numbers a(0),a(0)a(1),...,a(0)a(1)...a(n-1) give different remainder when divided by n.
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Re: Number Theory  
« Reply #1 on: Jul 7th, 2003, 3:56am »
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on Jul 7th, 2003, 12:10am, anonymous wrote:
the numbers a(0),a(0)a(1),...,a(0)a(1)...a(n-1)
What's happening here?  
I doubt it's multiplication, but concattenation also gives problems for n>10, unless you do the numbers in base n.
I really need a more precise description of what's happening.
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Re: Number Theory  
« Reply #2 on: Jul 7th, 2003, 4:10am »
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on Jul 7th, 2003, 3:56am, towr wrote:
What's happening here?

I haven't looked into the problem, but the notation used without any further explanation implies multiplication (imho).
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Re: Number Theory  
« Reply #3 on: Jul 7th, 2003, 4:16am »
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Yes, but then the answer is rather trivial, since you can't get different remainders for x and x*1, or x*0 and x*0*a(i)..a(j). So the answer is n=1,2.
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Re: Number Theory  
« Reply #4 on: Jul 7th, 2003, 4:49am »
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on Jul 7th, 2003, 4:16am, towr wrote:
So the answer is n=1,2.

How about n = 3?
a(j) = (1, 2, 0) works, doesn't it?
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Re: Number Theory  
« Reply #5 on: Jul 7th, 2003, 4:56am »
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hmm.. you're right..
So it just always has to start with 1 and end with 0..
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Re: Number Theory  
« Reply #6 on: Jul 7th, 2003, 5:10am »
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Yes, it's multiplication.
 
This is an old Bulgarian problem. The official problem text can be found on problems.math.umr.edu, but I've forgotten which year.
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Re: Number Theory  
« Reply #7 on: Jul 7th, 2003, 5:21am »
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so far I have
 
1: [0]
2: [1,0]
3: [1,2,0]
4: [1,3,2,0]
5: [1,2,4,3,0]
7: [1,3,4,6,2,5,0]
11: [1,7,6,4,5,2,3,8,10,9,0]
 
« Last Edit: Jul 7th, 2003, 7:05am by towr » IP Logged

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Re: Number Theory  
« Reply #8 on: Jul 7th, 2003, 7:39am »
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Hmm...
 
I think the solution is going to be {1,4,p where p is prime}. I can show that any composite number greater than 4 cannot do this, and I found an easy way to generate the sequence for any prime number. Now I just have to figure out why it works!
 
Think on this:
13: 1,2,8,10,11,9,12,3,6,4,5,7,0
 
I also think the question would be more intuitive if we used the numbers 1..N rather than 0..N-1.
 
UPDATE:
Yes, I think I can show that this method always works. So the numbers this works for are: {1, 4, all prime numbers}
« Last Edit: Jul 7th, 2003, 8:22am by James Fingas » IP Logged

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Re: Number Theory  
« Reply #9 on: Aug 29th, 2003, 1:13pm »
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on Jul 7th, 2003, 7:39am, James Fingas wrote:

 
Think on this:
13: 1,2,8,10,11,9,12,3,6,4,5,7,0
 
I also think the question would be more intuitive if we used the numbers 1..N rather than 0..N-1.
 
UPDATE:
Yes, I think I can show that this method always works. So the numbers this works for are: {1, 4, all prime numbers}

 
Yeap ... I found the same method and it's pretty easy to show that it always works.
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