thesilione's blog

Hard POI Problem
By thesilione, 12 years ago, In English

Could someone give me a hint for this problem?

http://main.edu.pl/en/archive/oi/2/jez

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12 years ago, hide # |
 
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In this problem we must use bfs on a graph, where each vertex is the remainder of the division by n.

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12 years ago, hide # |
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Why do you minusing it? Can't solve? That's nice problem

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    12 years ago, hide # ^ |
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    Oh, i thought that there is some condition like length of answer must be less than 1000. Of course, this problem is obvious: number 1111111....1111000000...000, with 9φ(n) of "1" and 30 "0" is an answer

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      12 years ago, hide # ^ |
       
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      Would you explain more?

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        12 years ago, hide # ^ |
         
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        Check out Euler Function and Euler`s theorem.

        We can reduce our problem to n so that gcd(n, 2) = gcd(n, 5) = 1. Now, we know that 10φ(n) - 1 is divisible on n, i.e. 99999...999 with φ(n) "9" is divisible on n. It`s easy to check that 1111111....1111000000...000, with 9φ(n) of "1" and 30 "0" is divisible on 10φ(n) - 1 so we are done.

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