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If n is an integer, and 3n+2 is even, prove that n is also even

  1. Feb 2, 2013 #1
    I am coming across hiccups in my proof process. I am given this problem - Prove: if n is an integer and 3n + 2 is even that n is also even. I have to apply a contrapositive proof to this problem. The form is then [itex]\neg[/itex]q therefore [itex]\neg[/itex]p .The problem becomes - if n is odd, prove that 3n+2 is even.

    Work:

    Prove - if n is odd, prove that 3n+2 is even.

    step 1 - if n is odd, n = 2k+1 for some integer k

    step 2 - 3n + 2 = 3(2k+1) + 2 = 6k + 5

    step 3 - This is my issue. A contrapositive proof for this problem would give not 'p', or, that 3n+2 is odd when n is odd. Do I now have to show that 6k + 5 is an odd number for any positive integer k? Or, should I just prove that 3n + 2 is odd when n is odd? If I take this route, could I choose another proof method, essentially having a 'proof within a proof'?
     
  2. jcsd
  3. Feb 2, 2013 #2

    pwsnafu

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    I'm assuming that's a typo.

    Probably not, but it's better to be safe. An odd number is an integer of form 2m+1. Find m and you are done.
     
  4. Feb 2, 2013 #3
    I am still not sure where I would take this proof. I apologize in advance as this is maybe the third proof I have done, and lack serious intution. How would I go about asserting that 6k + 5 is odd for any integer k? Would I do this:

    6k + 2 = 2m + 1

    m = 3k + 2

    ...however, this feels like circular logic. Does this mean that 6k+2 takes the form of an odd integer, and is therefore odd?
     
  5. Feb 2, 2013 #4

    rollingstein

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    What about this?

    3n+2=2k where k is some integer

    n=2(k-1)/3

    If n is an integer (given) it has to be even with 2 as a factor.
     
  6. Feb 2, 2013 #5

    pwsnafu

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    You need to stop being careless with your posting. It's "+5" not "+2".
    And yes. We prove something is odd by either
    1. Showing that it is equal to 2m+1 for some integer m, or
    2. Show the number is congruent to 1 (mod 2).
    And it's easy enough to show that those two statements amount to the same thing.

    That's not a contrapositive proof.
     
  7. Feb 2, 2013 #6

    rollingstein

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    Sorry. Didn't read that requirement. My bad.
     
  8. Feb 2, 2013 #7

    rollingstein

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    P1:If 3n+2 is even then n is also even

    P2:Contrapositive of P1: If n is odd then 3n + 2 is odd

    n=2k+1 where k=0,1,2,...

    3n+2=6k+3+2
    = 6k+5
    =6k+6-1
    =2(3k+3) - 1
    = even - 1
    = odd

    QED?
     
  9. Feb 13, 2013 #8

    ssd

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    3n+2 even
    =>3n even
    =>n even. This is the basic idea.
     
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