1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    None

    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

    RUber

    User Avatar
    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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Loading...