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Division Proof

  1. Feb 25, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that is m, n, and d are integers and d divides (m-n) then m mod d = n mod d.


    2. Relevant equations
    Quotient Remainder Theorem: Given any integer n and positive integer d, there exists unique integers q and r such that n=dq + r and 0[tex]\leq[/tex]r<d and n mod d = r.


    3. The attempt at a solution
    Proof: Let m, n, d [tex]\in[/tex] Z st d divides (m-n)
    [tex]\exists[/tex] k [tex]\in[/tex] Z st m=dk + r
    [tex]\exists[/tex] j [tex]\in[/tex] Z st n=dj + s
    m-n=(dk + r)-(dj + s)
    =dk+r-dj+s
    =d(k-j)+(r-s)

    Am I going along with the proof correctly? I don't know where to go from this point and would really appreciate some help.
     
  2. jcsd
  3. Feb 26, 2009 #2

    CompuChip

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    Science Advisor
    Homework Helper

    I'm not sure this is the easiest proof, but it looks correct.

    You have now shown that
    if m = r mod d, n = s mod d, then (m - n) mod d = (r - s)
    But you also know that d | (m - n) which you haven't used yet. So what does that tell you about (m - n) mod d?
     
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