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Thread: Number 1 Is Odd

  1. #1
    Super Member harpazo's Avatar
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    Number 1 Is Odd

    I know that 1 is an odd number but the reason is not clear to me. Why is 1 an odd number?
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  2. #2
    Junior Member Cervesa's Avatar
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    Re: Number 1 Is Odd

    a number is odd if it is not divisible by 2

    a number is odd if it can be written in the form $2k+1$ where $k$ is any integer
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  3. #3
    Super Member harpazo's Avatar
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    Re: Number 1 Is Odd

    Quote Originally Posted by Cervesa View Post
    a number is odd if it is not divisible by 2

    a number is odd if it can be written in the form $2k+1$ where $k$ is any integer
    Ok. Explain the 2k + 1 form.
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  4. #4
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    Re: Number 1 Is Odd

    Many questions in basic mathematics come down to the definitions!
    What definition of "odd number" are you using?

    I would say that an "odd number" is any integer that is not even. Or course, then I have to say what "even" means! An "even number" is an integer that is "evenly divisible" by 2: n is even if and only if n/2= k where k is an integer. That means that any even number is of the form n= 2k for some integer k.

    So what about odd numbers? Since an odd number is not even, if n is odd, n/2 cannot be an integer. Dividing n by 2 must gives some integer, k, plus a remainder. We can write n/2= k+ j/2 where j is the non-zero remainder.

    But the remainder when ever we divide one integer by another must be smaller than the divisor. When the divisor is 2, the only possible remainders are 0 and 1. Since j was to be "non-zero" it must be 1.

    That is, if n is odd, n/2= k+ 1/2. Multiplying both sides by 2, n= 2k+ 1.

    Any even number can be written in the form n= 2k for some integer k, any odd number can be written in the form n= 2k+ 1.

    (If k is an integer, so is k+ 1. If we write k'= k+1, then k= k'- 1 so n= 2(k'- 1)+ 1= 2k'- 2+ 1= 2k'- 1. Any odd number can also be written as "n= 2k- 1" just not the same k as before.)

    Another way of looking at this is that "even" and "odd" numbers alternate: given any even number, the number on either side, 1 more or 1 less, is odd.
    Last edited by HallsofIvy; Mar 10th 2019 at 07:49 AM.
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  5. #5
    Junior Member Cervesa's Avatar
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    Re: Number 1 Is Odd

    Quote Originally Posted by harpazo View Post
    Ok. Explain the 2k + 1 form.
    1 = 2(0) + 1

    zero is an integer
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    Super Member harpazo's Avatar
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    Re: Number 1 Is Odd

    Quote Originally Posted by Cervesa View Post
    1 = 2(0) + 1

    zero is an integer
    Why equate 2k + 1 to 1?
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  7. #7
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    Re: Number 1 Is Odd

    2(0) + 1 = 1
    2(1) + 1 = 3
    2(2) + 1 = 5
    2(3) + 1 = 7
    .....
    even + 1 = odd
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  8. #8
    Super Member harpazo's Avatar
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    Re: Number 1 Is Odd

    Quote Originally Posted by DenisB View Post
    2(0) + 1 = 1
    2(1) + 1 = 3
    2(2) + 1 = 5
    2(3) + 1 = 7
    .....
    even + 1 = odd
    Good.
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