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Math Help - [SOLVED] Problem from a section on Bezout's Identity

  1. #1
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    [SOLVED] Problem from a section on Bezout's Identity

    Find all solutions: 561x + 909y = 81

    Not just looking for answers, I don't understand the procedure. Any help is appreciated.
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  2. #2
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    Quote Originally Posted by Zennie View Post
    Find all solutions: 561x + 909y = 81

    Not just looking for answers, I don't understand the procedure. Any help is appreciated.

    Since (561,909)=3 and 3\mid 81 , there exists a solution in integers to the given equation :

    Lemma: If (m,n)=d \,\,\,and\,\,\,d\mid b\,,\,\,say\,\,\,b=kd then, after writing d=ma+nb\,,\,\,a,b\in\mathbb{Z} ,we get that
    m(ka)+n(kb)=kd=b , and mx+ny=b has an integer solution.

    This lemma is in fact an iff lemma.

    Tonio
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