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Proving Basic Theorem

  1. Sep 11, 2007 #1
    1. The problem statement, all variables and given/known data

    Prove a theorem using direct proof, mathematical induction, contraposition, or contradiction.

    2. Relevant equations

    "If m divides n, then m <= n."

    3. The attempt at a solution

    (a) Suppose m divides n, then m = nk for some integer k.

    (b) If k = 1, the m = n(1) = n.

    That show equality part. How do I now show inequality? I'm at a loss for the next step.
     
  2. jcsd
  3. Sep 11, 2007 #2

    HallsofIvy

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    If k is not 1, then, since it is a positive integer, k>1. What does that tell you?
     
  4. Sep 11, 2007 #3
    Given that, for k=1, m = n,

    then for k > 1 (or k + 1),

    m < n(k + 1).

    If I divide by k + 1,

    m / (k + 1) < n.

    Am I on the right track?
     
    Last edited by a moderator: Sep 11, 2007
  5. Sep 11, 2007 #4
    If m divides n (m<=n), then m*k=n and not m=n*k.

    If k=1, m=n, if k>1,

    mk=n => m<n.
     
  6. Sep 11, 2007 #5

    HallsofIvy

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    You're right! I missed that completely!
     
  7. Sep 11, 2007 #6
    Sleek, Thanks! I guess I got that twisted. Appreciate your helping me understand this logically.

    surferstrobe
     
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