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  • odd integer, squared , minus 1 ... the result is a multiple of _______

    From henhanna@gmail.com@21:1/5 to All on Wed Jul 20 08:33:01 2022
    Take an odd integer, square it, and subtract 1 ( One )... then
    the result is always a multiple of ____________ !




    ------- pls PLEASE wait a few days before posting answers or hints.





    (Sum of 2 squares)
    Pythagorean triple : (odd) ^ 2 + (odd) ^ 2 = (integer) ^ 2


    We've known about Squares ( and Pythagorean triples )
    for so long that.... When we learn something
    simple and new about them, it's like.... ___________

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  • From Edward Murphy@21:1/5 to henh...@gmail.com on Sun Jul 24 13:41:08 2022
    On 7/20/2022 8:33 AM, henh...@gmail.com wrote:

    Take an odd integer, square it, and subtract 1 ( One )... then
    the result is always a multiple of ____________ !

    [spoiler space]































    For any odd integer x:

    x = 2y + 1 (for some integer y)

    x^2 = 4y^2 + 4y + 1

    x^2 - 1 = 4y^2 + 4y
    = 4 * y * (y+1)

    Obviously it's divisible by 4, but also either y or y+1 is divisible by
    2 (if one is odd then the other is even), so x^2 - 1 is actually always divisible by 8.

    Sanity check:
    1^1 - 1 = 0 = 8 * 0
    3^1 - 1 = 8 = 8 * 1
    5^1 - 1 = 24 = 8 * 3
    7^1 - 1 = 48 = 8 * 6
    9^1 - 1 = 80 = 8 * 10
    and we can recognize the rightmost terms as
    y * (y+1) / 2
    for y = 0, 1, 2, 3, 4, etc.

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  • From Richard Heathfield@21:1/5 to Edward Murphy on Tue Jul 26 06:29:54 2022
    On 24/07/2022 9:41 pm, Edward Murphy wrote:
    On 7/20/2022 8:33 AM, henh...@gmail.com wrote:

    Take an odd integer,  square it, and subtract   1 ( One )...
    then
                    the result is always a multiple of
    ____________  !

    [spoiler space]

    Spoiler sans space: the answer is 1.

    TOO easy.

    --
    Richard Heathfield
    Email: rjh at cpax dot org dot uk
    "Usenet is a strange place" - dmr 29 July 1999
    Sig line 4 vacant - apply within

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