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The greatest common divisor of n integers

  1. Jul 29, 2012 #1
    1. The problem statement, all variables and given/known data
    define the gcd of a set of n integers, S={a_1...a_n}
    Prove that exists and can be written as
    q_1 a_1+...+q_n a_n for some integers, q_1....q_n

    2. Relevant equations

    Euclid's Algorithm?

    3. The attempt at a solution

    I have the statement that gcd(a_1....a_n) = min( gcd(a_i, a_j) i,j[itex]\in[/itex]{1...n})

    and then I know that I can say since any two numbers have a gcd, the gcd exists, and can be represented as a linear combination of the a_i and a_j that have the minimal gcd, but I don't know how to prove the first statement.
     
  2. jcsd
  3. Jul 30, 2012 #2
    I say this should be moved to the Precalculus help page...

    Anyhow... :) Well, for a start, your statement is NOT true.
    Consider {6,10,15}. Then gcd (6,10,15) = 1, but gcd(6,10) = 2, gcd(6,15) = 3, and gcd(10,15) = 10.

    Try solving it for n = 3. See if that gives you any ideas on how to approach the general case.
     
  4. Jul 30, 2012 #3
    This is easily done by induction (over the number of arguments) from gcd(a_1, a_2) = q_1 a_1 + q_2 a_2.
     
    Last edited: Jul 30, 2012
  5. Jul 30, 2012 #4
    You shouldn't merely give the answer - next time, try to lead the OP toward the correct solution. :)
     
  6. Jul 30, 2012 #5
    Thanks, i didnt catch the counterexample which set me off on the wrong track. Should be fine now
     
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