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Larson 4.1.8

  1. Jan 7, 2008 #1
    [SOLVED] larson 4.1.8

    1. The problem statement, all variables and given/known data
    Let N be the number which when expressed in decimal notation consists of 91 ones:

    1111...1111 = N

    Prove that N is a composite number.
    2. Relevant equations



    3. The attempt at a solution
    If N had an even number n of ones we could use the fact that

    [tex]\sum_{i=0}^n x^i = (1+x)(x+x^3+x^5+...+x^{n-1}) = N [/tex]

    evaluated at x=10. I tried doing lots of similar tricks for the odd case but nothing seems to factor completely.
     
  2. jcsd
  3. Jan 7, 2008 #2

    morphism

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    Science Advisor
    Homework Helper

    Since 91=13*7, then 10^91 - 1 = 0 (mod 10^7 - 1) (and 10^91 - 1 = 0 (mod 10^13 - 1)) - prove this. And since 9N = 10^91 - 1, it follows that N is divisible by (10^7 - 1)/9 (and (10^13 - 1)/9).
     
    Last edited: Jan 7, 2008
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