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Math Help - Divisibility of 0

  1. #1
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    Divisibility of 0

    Hey all, any help on the following proofs would be appreciated:

    (i) 0 is divisible by every integer

    (ii) If m is an integer not equal to 0, then m is not divisible by 0.
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  2. #2
    Senior Member MacstersUndead's Avatar
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    Quote Originally Posted by jstarks44444 View Post
    Hey all, any help on the following proofs would be appreciated:

    (i) 0 is divisible by every integer

    (ii) If m is an integer not equal to 0, then m is not divisible by 0.
    --
    For (i) use the fact that 0 = 0*a, for any integer a

    For (ii), suppose not. Suppose that a/0 = b for a a non-zero integer and b some real number. try to find a contradiction, using the axiom 0 =! 1
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by jstarks44444 View Post
    Hey all, any help on the following proofs would be appreciated:

    (i) 0 is divisible by every integer

    (ii) If m is an integer not equal to 0, then m is not divisible by 0.
    (i) For all n\in\mathbb{Z}, 0=0\cdot n\implies n\mid 0.

    (ii) Suppose 0\mid m. Then \exists\,k\in\mathbb{Z}: m = k\cdot 0. So what can be said about m?
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  4. #4
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    This is another example of a very basic principle- proofs follow from the precise words of definitions.

    "m is divisible by n" means that there exist some integer, k, such that m= nk.

    If m= 0, is there an integer, k, such that nk= 0?

    If m is NOT 0, does there exist an integer, k, such that 0k= m?
    Last edited by HallsofIvy; February 11th 2011 at 04:52 AM.
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