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Discrete mathematics induction

  1. Nov 11, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that for all integers a >= 1, a^n - 1 is divisible by a - 1 for all n >= 1.

    2. Relevant equations

    None.

    3. The attempt at a solution

    Proof - Let P(n): a^n - 1 is divisible by a - 1, then

    P(1): a^1 - 1 is divisible by a - 1 is TRUE since a^1 - 1 = a - 1, and a - 1 = (1)(a - 1)

    Suppose that P(k): a^k - 1 is divisible by a - 1 is TRUE. Then, there exists an integer q such that a^k - 1 = (a - 1)q or a^k = (a-1)q + 1

    Now consider a^(k+1) - 1 = a(a^k) - 1
    = a((a - 1)q + 1) - 1
    = (a)(a - 1)q + a - 1
    = (a - 1)aq + (a - 1)
    = (a - 1)(aq + 1)

    Thus, a^(k+1) - 1 is divisible by a - 1 whenever a^k - 1 is divisible by a - 1. Therefore, a^n - 1 is divisible by a -1 for all n >= 1.


    The problem I'm having is I don't think I proved that for all a >= 1.
     
  2. jcsd
  3. Nov 11, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    No, there is no requirement that a- 1 be greater than 0 so there is no requirement that a be greater than 1.
     
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