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Math Help - Even numbers and odd numbers..

  1. #1
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    Even numbers and odd numbers..

    If a,b,c and d are four positive real numbers such that the sum of a,b and c is even and the sum of b,c and d is odd, then a^2 - d^2 is neccesarily

    Please explain clearly...
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  2. #2
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    Re: Even numbers and odd numbers..

    Quote Originally Posted by kohila View Post
    If a,b,c and d are four positive real numbers such that the sum of a,b and c is even and the sum of b,c and d is odd, then a^2 - d^2 is neccesarily
    I think that you mean "a,b,c and d are four positive integers" otherwise no conclusion is possible.
    If that is the case then you can show a-d=2K-1 which is odd and a+d=2J+1-2b-2c which is also odd.

    But a^2-b^2=(a-b)(a+b) SO?
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  3. #3
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    Re: Even numbers and odd numbers..

    Hello, kohila!

    I have a primitive solution with brute-force Listing.


    \text{If }a,b,c,d\text{ are four positive integers such that }a+b+c\text{ is even}
    \text{and }b+c+d\text{ odd, then }a^2 - d^2\text{ is neccesarily }\_\_\_.

    Let \begin{Bmatrix} e &=& \text{even} \\ o &=& \text{odd}\end{Bmatrix}

    Since a+b+c is even, either (1) all are even or (2) exactly one is even.

    Hence, there are four cases to consider:
    . . (1) a,b,c are even.
    . . (2) a even, b,c odd.
    . . (3) b even, a,c odd.
    . . (4) c even, a,b odd.


    (1)\;[a\;b\;c\;d] \,=\,[e\;e\;e\;\_\,]
    . . .Since b+c+d is odd, d must be odd.
    . . .Hence: a is even, d is odd.

    (2)\;[a\;b\;c\;d] \,=\,[e\;o\;o\;\_\,]
    . . .Since b+c+d is odd, d must be odd.
    . . .Hence, a is even, d is odd.

    (3)\;[a\;b\;c\;d] \,=\,[o\;e\;o\;\_\,]
    . . .Since b+c+d is odd, d must be even.
    . . .Hence, a is odd, d is even.

    (4)\;[a\;b\;c\;d] \,=\,[o\;o\;e\;\_\,]
    . . .Since b+c+d is odd, d must be even.
    . . .Hence, a is odd, d is even.


    In all cases, a and d have opposite parity.
    Hence, their squares will have opposite parity.

    The difference between two numbers of opposite parity is odd.

    Thanks from kohila
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  4. #4
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    Re: Even numbers and odd numbers..

    Quote Originally Posted by kohila View Post
    If a,b,c and d are four positive real numbers such that the sum of a,b and c is even and the sum of b,c and d is odd, then a^2 - d^2 is neccesarily

    Please explain clearly...
    Necessarily...odd? Also, a,b,c,d should be integers, otherwise we cannot claim anything.

    We have
    a + b + c \equiv 0 (\mod 2)
    b + c + d \equiv 1 (\mod 2)

    If we subtract the second equation from the first, we get a-d \equiv 1 (\mod 2), therefore a-d is odd. This implies that a+d is also odd, since a+d = (a-d) + 2d. Therefore (a-d)(a+d) = a^2 - d^2 must be odd.
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  5. #5
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    Re: Even numbers and odd numbers..

    Thanks... I can get it from your explanation.
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  6. #6
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    Re: Even numbers and odd numbers..

    Much thanks for ur deep core conceptual clarification...
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