## lgbasallote 3 years ago Prove that the product of two odd integers is odd

1. lgbasallote

i suppose the first step is to let the integers be a and b

2. ParthKohli

I'd probably use proof by contrapositive.

3. klimenkov

Let the first one be \(2n+1\), the second - \(2k+1\). Multiply -\((2n+1)(2k+1)=2(2nk+k+n)+1\) - odd.

note thta the product of two even is even

6. ParthKohli

...or the direct proof suggested by @klimenkov

7. lgbasallote

no one spoil the steps yer......

8. lgbasallote

let the asker finish what he's typing first...im just too lagged...

9. lgbasallote

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

10. lgbasallote

but this is just direct proof and too simple and dull....how to prove by contradiction?

11. lgbasallote

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?

12. SmoothMath

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.

13. lgbasallote

care to elaborate?

\[m*n=(2k)(2x)=2(2kx)\]

the product of two non odd integers is non odd

16. SmoothMath

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.

17. estudier

(2k-1)(2k+1) = 4k^2 - 1

18. lgbasallote

isn't that what i just said?

19. SmoothMath

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."

20. estudier

It's a slightly neater version of what Klimenkov said...

21. lgbasallote

22. estudier

:-)

23. SmoothMath

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.

24. lgbasallote

im not in college...i'm a sophomore high school...

25. estudier

Number theory proofs are allowed to be a bit more casual than , say , analytic proofs...

26. SmoothMath

Haha I know, buddy. And it's incredible that you can write proofs like this already. You're a champion.

27. lgbasallote

are analytic proofs the ones i do and prefer?

28. estudier

Analytic is fancy calculus, real numers, continuity, blah.....

29. lgbasallote

so what am i doing?

30. SmoothMath

Estudier, I was never allowed to do any purely algebraic proofs. I was required formal paragraph proofs pretty much 100% of the time.

31. SmoothMath

But my study was certainly skewed towards the analytic side.

32. lgbasallote

like those geometric proofs with all the tables?

33. estudier

(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....

CALCULUS LIMIT,AND CONT. PROOFS ARE TYPICAL ANALYTICAL PROOFS

35. SmoothMath

Ew... I totally disagree. You've made so many assumptions there.

36. lgbasallote

@estudier you love skipping steps a lot, don't you

37. estudier

What, like 2k-1 is odd, u mean?

38. lgbasallote

like proving square of even is even

39. estudier

or that 4k^2 is even...

40. SmoothMath

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

41. lgbasallote

it's the principle you used for (2k)^2 - 1 after all

42. lgbasallote

i think "square of even is even" is what they call conjecture

43. SmoothMath

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.

44. estudier

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....

45. lgbasallote

like i said...im just in primary school...not in number theory class

46. estudier

Ah well, this is all a question of what u are allowed to assume....

47. SmoothMath

... 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

WHAT DO LITERALLY MEAN YOU ARE IN PRIMARY

49. lgbasallote

primary school -> the educational level before high school and after preschool

50. SmoothMath

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.

are you in primary

52. SmoothMath

product is odd*

53. lgbasallote

yes

why are you doing discreet mathematics which i am doing at college

55. SmoothMath

Because LGBA is a boss.

56. lgbasallote

i believe that education belongs to no level

lol he really is i thought you are a teacher at some school,

58. estudier

I'm with Klimenkov (but we are not in primary school, I guess)

59. lgbasallote

^he's smart

60. lgbasallote

i meant the one above estudier.......

62. SmoothMath

I'm a teacher, if that's what you were asking. I teach algebra 2 and geometry.

@klimenkov and @estudier @lgbasallote

prodigees

65. lgbasallote

i'm actually a liberal arts teacher in primary school. i was just kidding about that primary school thing

lol i was just about to say you shuld take the IMO

67. lgbasallote

IMO?

but it is still true ,you are in primary

69. klimenkov

I'm 20. I just like math.

70. lgbasallote

somehow yes

73. lgbasallote

no thanks....as you can see from my picture...i don't really like math....

lol you guys are funny,if you hated maths you wont be on openstudy is that a true proposition

75. lgbasallote

openstudy is not just about math

76. SmoothMath

Nay.

77. lgbasallote

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

p=you hate math q=you are not on OS what is the contrapositive

79. estudier

por q or p xor q or?

not clear,i dont get it @estudier

81. estudier

Just me not being funny....:)

\[¬(p→q) \]expand

83. estudier

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

ohh thats true but i was talking abut \[¬(p→q) ≡ p ⋀ (¬q)\]

85. estudier

p->q is not p or t so not that...

86. estudier

q, I mean, not t