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Divisibility Proof (Abstract Algebra)

  1. Mar 25, 2009 #1
    1. The problem statement, all variables and given/known data

    let a belong to N and x,r belong to Z use the definition of divisibility along with the axioms of Integers to prove that IF 5|a and 15|(2ax+r) then 5|r

    2. Relevant equations How do I continue the proof??



    3. The attempt at a solution So I have: let a belong to N and x,r belong to Z. Assume 5|a and 15|(2ax+r). Then there is s,t belonging to Z such that a=5s and (2ax+r)=3(5T).
     
  2. jcsd
  3. Mar 25, 2009 #2

    jambaugh

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    How does the definition apply to 5|r? Can you get the form required to show it?
     
  4. Mar 25, 2009 #3
    It is an if then proof with 5|r being what you are trying to prove.
     
  5. Mar 25, 2009 #4
    Use the fact that 5|a to substitute.
     
  6. Mar 25, 2009 #5

    jambaugh

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    Yes of course but you must end up with your next to last step being the defining form of 5|r and the last step being "thus 5|r".

    You almost have it. Play with what you have and what you need and see if you can connect the two. I can tell you but the point of you going through the proof is you going through the proof.
     
  7. Mar 25, 2009 #6
    I now have this so
    r=15t-2ax=
    15t-2(5s)x=
    5(3T)-5(2s)x=
    5(3t-2sx)=r
    since 3t-2sx belongs to Z we have 5|r

    does that make sense??
     
  8. Mar 25, 2009 #7
    Looks good.
     
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