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Homework Help: GCD Number Theory Problem

  1. Jan 7, 2015 #1
    1. The problem statement, all variables and given/known data
    if 1 = gcd(a,b), show that gcd(ac,b) = gcd(c,b)

    2. Relevant equations

    3. The attempt at a solution
    My attempt at a solution:

    Let d = gcd(ac,b),
    Let g = gcd(c,b),
    I want to show that g|d and that d|g. I then went on to make a bunch of circular writing and get nowhere... I set up things like:

    1 = ax + by
    d = acw + bz --> 1 = (ac/d)w + (b/d)z
    g = cn + bm --> 1 = (c/d)n + (b/d)m

    Is my approach here a solid method? I can't think of any other way to show that gcd(ac,b) = gcd(c,b) besides assigning each of them a value and showing that they divide eachother.
  2. jcsd
  3. Jan 7, 2015 #2


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    Homework Helper

    That seems to be a reliable method.
    I prefer to work with factor sets:
    Let A be the unique prime factor set of a, B be the unique prime factor set of b, and C be the unique prime factor set of c.
    ##A\cap B =\{1\}##
    ##AC=A \cup C##
    ##AC\cap B= (A \cup C) \cap B = (A\cap B) \cup (C \cap B) = 1 \cup (C \cap B) ##
    A similar argument should apply with repeated prime factors and combinations to find the gcd.
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