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Math Help - Abstract Algebra

  1. #1
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    Abstract Algebra

    Show that Z (integers) is generated by 5 and 7.

    I can sit and play with those two numbers to get all Z, but can't come up with a formula. Can someone show how to do this?

    Thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Abstract Algebra

    Quote Originally Posted by jzellt View Post
    Show that Z (integers) is generated by 5 and 7.
    For all k\in\mathbb{Z} we have k=k\cdot 1=k\left(3\cdot 5+(-2)\cdot 7\right)=(3k)\cdot 5+(-2k)\cdot 7.
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  3. #3
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    Re: Abstract Algebra

    Thanks! That's brilliant!
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Smile Re: Abstract Algebra

    Quote Originally Posted by jzellt View Post
    Thanks! That's brilliant!
    Not very brilliant . It is a particular case of the Bezout equallity.
    Last edited by FernandoRevilla; September 20th 2012 at 01:45 AM.
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  5. #5
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    Re: Abstract Algebra

    the subgroup of Z generated by the integers k and m is: <gcd(k,m)>. if two numbers are co-prime, they generate all of Z, since 1 generates Z (Z is cyclic).
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