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Math Help - Finding inverse inside a field

  1. #1
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    Finding inverse inside a field

    I am using euclidean algorithm to find if 7 inverse exists in mod12

    I ended up with

    1=3(12)-5(7)

    So the inverse of 7 is -5

    and this is what the instructor wrote in the notes but he went from

    7^-1=5(mod12)

    to

    7^-1=7(mod12)

    what happened here and how did he end up with 7 inverse being 7 in the field of mod12????
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  2. #2
    MHF Contributor
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    Re: Finding inverse inside a field

    We have 7\equiv-5\pmod{12} because 7 - (-5) is a multiple of 12 (or, 7 = -5 + 12). On the other hand, 5\not\equiv-5\pmod{12}. So indeed, 7\cdot 7=49=4\cdot12+1\equiv1\pmod{12}.
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