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lgbasallote Group Title

Prove that the product of two odd integers is odd

  • one year ago
  • one year ago

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  1. lgbasallote Group Title
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    i suppose the first step is to let the integers be a and b

    • one year ago
  2. ParthKohli Group Title
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    I'd probably use proof by contrapositive.

    • one year ago
  3. klimenkov Group Title
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    Let the first one be \(2n+1\), the second - \(2k+1\). Multiply -\((2n+1)(2k+1)=2(2nk+k+n)+1\) - odd.

    • one year ago
  4. Jonask Group Title
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    note thta the product of two even is even

    • one year ago
  5. Jonask Group Title
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    contrad

    • one year ago
  6. ParthKohli Group Title
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    ...or the direct proof suggested by @klimenkov

    • one year ago
  7. lgbasallote Group Title
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    no one spoil the steps yer......

    • one year ago
  8. lgbasallote Group Title
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    let the asker finish what he's typing first...im just too lagged...

    • one year ago
  9. lgbasallote Group Title
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    anyway... so i let m = 2k + 1 n = 2x + 1 now... solving for the product mn = (2k + 1)(2x + 1) mn = 4kx + 2k + 2x + 1 then... mn = 2(2kx + k + x) + 1 so it's odd

    • one year ago
  10. lgbasallote Group Title
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    but this is just direct proof and too simple and dull....how to prove by contradiction?

    • one year ago
  11. lgbasallote Group Title
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    i suppose i do mn = 2y where m and n are odd (2k + 1)(2x+1) = 2y 4kx + 2k + 2x + 1 = 2y then... 4kx + 2k + 2x - 2y = -1 2(2kx + k + x - y) = -1 then... 2kx + k + x - y = -1/2 since k, x and y are integers, the right side should also be an integer. thus, contradiction. is that how it's done?

    • one year ago
  12. SmoothMath Group Title
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    That's how it's done =) The only thing to point out is that this is not a formal proof. To formalize it, you would need to clearly state all of your assumptions, plus give a reasoning for all of your steps. Then at the end, you would have to explicitly state what contradiction you have uncovered.

    • one year ago
  13. lgbasallote Group Title
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    care to elaborate?

    • one year ago
  14. Jonask Group Title
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    \[m*n=(2k)(2x)=2(2kx)\]

    • one year ago
  15. Jonask Group Title
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    the product of two non odd integers is non odd

    • one year ago
  16. SmoothMath Group Title
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    In this case, the contradiction is that k,x, and y are all integers. The product of integers is an integer, therefore 2kx is an integer. The sum or difference of integers is an integer, therefore 2kx + k + x -y is an integer. You conclude that 2kx + k +x -y = -1/2 So that is a contradiction.

    • one year ago
  17. estudier Group Title
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    (2k-1)(2k+1) = 4k^2 - 1

    • one year ago
  18. lgbasallote Group Title
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    isn't that what i just said?

    • one year ago
  19. SmoothMath Group Title
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    And the part where I say to state your assumptions, it would simply be what you have there, except you would say "Assume that m and n are odd numbers such that their product is even. M is odd, therefore m = 2k+1 for some integer k. N is odd, therefore n = 2x+1 for some integer x. Their product is even, therefore mn = 2y for some integer y."

    • one year ago
  20. estudier Group Title
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    It's a slightly neater version of what Klimenkov said...

    • one year ago
  21. lgbasallote Group Title
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    @estudier please don;t mix your complicated proofs here......

    • one year ago
  22. estudier Group Title
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    :-)

    • one year ago
  23. SmoothMath Group Title
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    LGBA, I'm only nitpicking in the way a college professor would nitpick. Your proof is great, it's just not formal or rigorous the way it is required to be in a college course.

    • one year ago
  24. lgbasallote Group Title
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    im not in college...i'm a sophomore high school...

    • one year ago
  25. estudier Group Title
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    Number theory proofs are allowed to be a bit more casual than , say , analytic proofs...

    • one year ago
  26. SmoothMath Group Title
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    Haha I know, buddy. And it's incredible that you can write proofs like this already. You're a champion.

    • one year ago
  27. lgbasallote Group Title
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    are analytic proofs the ones i do and prefer?

    • one year ago
  28. estudier Group Title
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    Analytic is fancy calculus, real numers, continuity, blah.....

    • one year ago
  29. lgbasallote Group Title
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    so what am i doing?

    • one year ago
  30. SmoothMath Group Title
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    Estudier, I was never allowed to do any purely algebraic proofs. I was required formal paragraph proofs pretty much 100% of the time.

    • one year ago
  31. SmoothMath Group Title
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    But my study was certainly skewed towards the analytic side.

    • one year ago
  32. lgbasallote Group Title
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    like those geometric proofs with all the tables?

    • one year ago
  33. estudier Group Title
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    (2k-1)(2k+1) = 4k^2 - 1 QED is an acceptable proof. You can dress it up with some words and put in a paragraph if u like....

    • one year ago
  34. Jonask Group Title
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    CALCULUS LIMIT,AND CONT. PROOFS ARE TYPICAL ANALYTICAL PROOFS

    • one year ago
  35. SmoothMath Group Title
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    Ew... I totally disagree. You've made so many assumptions there.

    • one year ago
  36. lgbasallote Group Title
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    @estudier you love skipping steps a lot, don't you

    • one year ago
  37. estudier Group Title
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    What, like 2k-1 is odd, u mean?

    • one year ago
  38. lgbasallote Group Title
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    like proving square of even is even

    • one year ago
  39. estudier Group Title
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    or that 4k^2 is even...

    • one year ago
  40. SmoothMath Group Title
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    For one thing, you'd like for it to be true with any two odd integers. Your proof assumes that the integers are two apart. 2k+1 and 2k-1

    • one year ago
  41. lgbasallote Group Title
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    it's the principle you used for (2k)^2 - 1 after all

    • one year ago
  42. lgbasallote Group Title
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    i think "square of even is even" is what they call conjecture

    • one year ago
  43. SmoothMath Group Title
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    And I mean... call me old fashioned, but I really hate that you're not stating your assumptions at all. You have 1 line. Gawd that gives me the jibblies in a real bad way.

    • one year ago
  44. estudier Group Title
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    In number theory class, I can assure u that will not be required to prove that odd numbers are of the form 2k plus/minus 1 or that odd numbers are 2 apart....

    • one year ago
  45. lgbasallote Group Title
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    like i said...im just in primary school...not in number theory class

    • one year ago
  46. estudier Group Title
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    Ah well, this is all a question of what u are allowed to assume....

    • one year ago
  47. SmoothMath Group Title
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    ... Estudier, you've proved, for example, that the product of 5 and 7 is even, because you could choose k=6. However, you haven't proved that the product of 5 and 9 is even, because you can not choose a k such that (2k+1)(2k-1) = 5*9

    • one year ago
  48. Jonask Group Title
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    WHAT DO LITERALLY MEAN YOU ARE IN PRIMARY

    • one year ago
  49. lgbasallote Group Title
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    primary school -> the educational level before high school and after preschool

    • one year ago
  50. SmoothMath Group Title
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    Do you see my point, Estudier? The task is to prove that the product of ANY two odd integers is even. You've only proven that the product of consecutive odd integers is even.

    • one year ago
  51. Jonask Group Title
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    are you in primary

    • one year ago
  52. SmoothMath Group Title
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    product is odd*

    • one year ago
  53. lgbasallote Group Title
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    yes

    • one year ago
  54. Jonask Group Title
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    why are you doing discreet mathematics which i am doing at college

    • one year ago
  55. SmoothMath Group Title
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    Because LGBA is a boss.

    • one year ago
  56. lgbasallote Group Title
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    i believe that education belongs to no level

    • one year ago
  57. Jonask Group Title
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    lol he really is i thought you are a teacher at some school,

    • one year ago
  58. estudier Group Title
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    I'm with Klimenkov (but we are not in primary school, I guess)

    • one year ago
  59. lgbasallote Group Title
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    ^he's smart

    • one year ago
  60. lgbasallote Group Title
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    i meant the one above estudier.......

    • one year ago
  61. Jonask Group Title
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    what grade are you guys

    • one year ago
  62. SmoothMath Group Title
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    I'm a teacher, if that's what you were asking. I teach algebra 2 and geometry.

    • one year ago
  63. Jonask Group Title
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    @klimenkov and @estudier @lgbasallote

    • one year ago
  64. Jonask Group Title
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    prodigees

    • one year ago
  65. lgbasallote Group Title
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    i'm actually a liberal arts teacher in primary school. i was just kidding about that primary school thing

    • one year ago
  66. Jonask Group Title
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    lol i was just about to say you shuld take the IMO

    • one year ago
  67. lgbasallote Group Title
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    IMO?

    • one year ago
  68. Jonask Group Title
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    but it is still true ,you are in primary

    • one year ago
  69. klimenkov Group Title
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    I'm 20. I just like math.

    • one year ago
  70. lgbasallote Group Title
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    somehow yes

    • one year ago
  71. Jonask Group Title
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    IMO-international math olympiad

    • one year ago
  72. Jonask Group Title
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    IMO-international math olympiad

    • one year ago
  73. lgbasallote Group Title
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    no thanks....as you can see from my picture...i don't really like math....

    • one year ago
  74. Jonask Group Title
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    lol you guys are funny,if you hated maths you wont be on openstudy is that a true proposition

    • one year ago
  75. lgbasallote Group Title
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    openstudy is not just about math

    • one year ago
  76. SmoothMath Group Title
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    Nay.

    • one year ago
  77. lgbasallote Group Title
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    p: i like math q: i'm on openstudy if p then q since p is F and q is T..the implication holds true

    • one year ago
  78. Jonask Group Title
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    p=you hate math q=you are not on OS what is the contrapositive

    • one year ago
  79. estudier Group Title
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    por q or p xor q or?

    • one year ago
  80. Jonask Group Title
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    not clear,i dont get it @estudier

    • one year ago
  81. estudier Group Title
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    Just me not being funny....:)

    • one year ago
  82. Jonask Group Title
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    \[¬(p→q) \]expand

    • one year ago
  83. estudier Group Title
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    x,y odd -> x-1, y-1 even (I don't have to prove that as well, do I?) x-1 = 2p, y-1 =2q -> x = 2p+1, y =2q+1 (2p+1)(2q+1) = 4pq +2p +2q +1 QED

    • one year ago
  84. Jonask Group Title
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    ohh thats true but i was talking abut \[¬(p→q) ≡ p ⋀ (¬q)\]

    • one year ago
  85. estudier Group Title
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    p->q is not p or t so not that...

    • one year ago
  86. estudier Group Title
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    q, I mean, not t

    • one year ago
  87. Jonask Group Title
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    yes

    • one year ago
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