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Ring theory question

  1. Jan 26, 2008 #1
    [SOLVED] ring theory question

    1. The problem statement, all variables and given/known data
    My book says that "if a is an integer relatively prime to n, then the coset a +nZ of nZ containing contains an integer b < n and relatively prime to n."

    If 0< a < n, this is obvious. If a > n or 0 > a, I do not see why that statement is true.


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 26, 2008 #2
    Actually, I do see why it is true. By the division algorithm, we have

    a=qn+r, where 0 <= r < n

    If gcd(n,r)=d>1, then d must divide a.

    So, b=r.
     
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