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Math Help - Divisibility 11

  1. #1
    Sea
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    Divisibility 11

    Show that: a, k\in \mathbf{Z},


    a=2k \Leftrightarrow a-2\| \frac{a}{2}\| = 0


    .............................................

    Symbol...
    Example:

    \| 5,3\|=5

    \| -2,2\|=-3
    Last edited by Sea; December 20th 2008 at 12:56 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Sea View Post
    Show that: a, k\in \mathbf{Z},


    a=2k \Leftrightarrow a-2\| \frac{a}{2}\| = 0


    .............................................

    Symbol...
    Example:

    \| 5,3\|=5

    \| -2,2\|=-3
    In more familiar notation:

    Show that: a, k\in \mathbb{Z},


    a=2k \Leftrightarrow a-2 \left\lfloor \frac{a}{2}\right\rfloor = 0

    where \lfloor x \rfloor is the floor function, the greatest integer less than x.

    CB
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    In more familiar notation:

    Show that: a, k\in \mathbb{Z},


    a=2k \Leftrightarrow a-2 \left\lfloor \frac{a}{2}\right\rfloor = 0

    where \lfloor x \rfloor is the floor function, the greatest integer less than x.

    CB
    Suppose a=2k for some k \in \mathbb{Z} then:

     <br />
a-2\left\lfloor \frac{a}{2}\right\rfloor=a-2 \lfloor k \rfloor=a-2k=0<br />

    Which proves

    a=2k \Rightarrow a-2 \left\lfloor \frac{a}{2}\right\rfloor = 0

    To complete the proof assume that: a-2 \left\lfloor \frac{a}{2}\right\rfloor = 0 and show that this implies that there exists a k \in \mathbb{Z} such that a=2k

    CB
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  4. #4
    Sea
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    I didn't know this words means in english...I'm sorry for this and I thank you so much... for answering my question after all...



    Thanks a million......
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