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Math Help - [SOLVED] Coprime

  1. #1
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    [SOLVED] Coprime

    Hi, i'm having trouble with the following questions;

    Let a, b be non-zero integers.

    a) Show that if there are x,y in Z (integers) with ax + by =1, then a and b are coprime.

    b) Show that the integers a/(a,b) and b/(a,b) are coprime.

    Thanks for you help.
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  2. #2
    MHF Contributor red_dog's Avatar
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    1) Let d such as d|a and d|b

    Then d|ax+by\Rightarrow d|1\Rightarrow d=1
    So, a and b are coprime.

    2) Let d=(a,b)

    Then a=da_1, \ b=db_1 with (a_1,b_1)=1

    But \frac{a}{d}=a_1, \ \frac{b}{d}=b_1, so \frac{a}{d}, \ \frac{b}{d} are coprime.
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