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Math Help - Langrange theorem problem....

  1. #1
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    Post Langrange theorem problem....

    Find [ Z_40 : <[12], [20]> ]?

    I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.

    Thanks for the help!
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Re: Langrange theorem problem....

    Quote Originally Posted by vedicmath View Post
    Find [ Z_40 : <[12], [20]> ]?

    I know it has something to do with the gcd of 12 and 20 which is 4 but I do not know how to proceed with it and simply it with proper steps.

    Thanks for the help!
    So, it is "well known" that any subgroup of a cyclic group is cyclic. As you have pointed out, gcd(12, 20)=4. This means that the subgroup generated by 12 and 20 is equal to the subgroup generated by 4, \langle 12, 20\rangle=\langle 4\rangle (this should be in your notes, and is the crux of the argument - if you do not know it, learn it!). So, what is the index of \langle 4\rangle in \mathbb{Z}_4? Divide 40 by this order to get your answer (why?).
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