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Algebra Question: 2r - 2s - 1 = 2(r-s-1) + 1 (?)

  1. Jul 15, 2012 #1
    1. The problem statement, all variables and given/known data
    Well, the problem is a discrete math problem to prove that the difference of an even integer and an odd integer is an odd integer.

    2. Relevant equations
    If you let m = an even integer, and n = an odd integer then m = 2r for some integer r, while n = 2s + 1 for some integer s.

    3. The attempt at a solution
    Then
    m - n = 2r - (2s + 1) = 2r - 2s - 1

    at the next step, the answer in the book shows
    2(r-s-1)+1, from there I can follow the rest of the steps to show that an integer is odd but I'm confused as to where this extra one is coming from. I know it's just a basic factoring question, but I'm a little rusty, maybe somebody could help me.. thanks.
     
  2. jcsd
  3. Jul 15, 2012 #2

    Mark44

    Staff: Mentor

    2r - 2s - 1 = 2r - 2s - 1 - 1 + 1 = 2r - 2s - 2 + 1 = 2(r - s - 1) + 1

    In the second step above, I added 0, in the form of -1 + 1. You can always add 0 to an expression to get an identically equal expression.
     
  4. Jul 15, 2012 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    -1 = -2 + 1

    rgv
     
  5. Jul 15, 2012 #4
    Aha, can't believe I missed such an easy solution, guess it's good to humble one down, thanks for the help mates... I'm new to this forum is there anything like +rep or likes that I can give?
     
  6. Jul 15, 2012 #5
    What am I missing? Why do you have to put it as 2(r-s-1)+1 to show it is odd? Why isn't it acceptable (and slightly simpler) just to put it as 2(r-s)-1 ?
     
  7. Jul 15, 2012 #6
    Unfortunately no, you can't give rep; but I know some would have a whole lot more than the rest of us!

    It's probably because odd numbers are usually written in the form 2(some letter or quantity) + 1 instead of 2(some letter or quantity) - 1.
     
  8. Jul 15, 2012 #7
    Yep, that's what I assumed. Seems a bit daft to me. :yuck:
     
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