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Homework Help: Proof involving divisibilty

  1. May 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Show n4 - 1 is divisible by 5 when n is not divisible by 5.



    2. Relevant equations



    3. The attempt at a solution

    My thought here is proof by cases, one case being where n is divisible by 5 and the other case saying n is not divisible by 5. I am just not totally sure how to implement this strategy.
     
  2. jcsd
  3. May 2, 2010 #2
    If you split your second case a bit more that will do fine.
     
  4. May 3, 2010 #3

    HallsofIvy

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    Science Advisor

    If n is not divisible by 5, then it is of the form n= 5k+ 1, n= 5k+ 2, n= 5k+ 3, or n= 5k+ 4.

    Or you can consider n= 5k- 2, n= 5k- 1, n= 5k+ 1, and n= 5k+ 2 if that makes the calculations simpler.
     
  5. May 4, 2010 #4
    If we would to spent a little more time to think through , we will realise that the determining factor will then lies in [tex]5k \pm m[/tex] such the [tex]m^{4}\equiv 1 (mod5)[/tex] with application with binomial theorem.
     
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