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Proving 5 integers to be pairwise relatively prime

  1. Sep 3, 2016 #1
    1. The problem statement, all variables and given/known data
    Let n be an integer. Prove that the integers 6n-1, 6n+1, 6n+2, 6n+3, and 6n+5 are pairwise relatively prime.

    2. Relevant equations


    3. The attempt at a solution
    I tried to prove that the first two integers in the list are relatively prime.

    (6n-1)-(6n+1)=1 (trying to eliminate the n variable)
    6n-1-6n-1=1
    -2=1, which is obviously not true.
    Not sure where to go from here. Is there another way to prove that two integers are relatively prime?
     
  2. jcsd
  3. Sep 3, 2016 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    You can make at least some progress by noting that ##O = 6n-1## is an odd number, and then arguing that for any odd number ##O##, the pair ##O, O+2## are relatively prime. That also takes care of the pairs ##6n+1, 6n+3##, ##6n+3, 6n+5##. You also have that ##E = 6n+2## is an even number, and can argue that ##E## and ##E+1## are relatively prime, so that takes care of ##6n+2, 6n+3##. That leaves a few more similar pairs to check.

    As for showing relative primeness, you just need to show that any factor of one of the numbers fails to be a factor of the other (except for 1, of course). You may be able to do it by contradiction.
     
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