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Math Help - Divisibility (gcd) 2

  1. #1
    Sea
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    Divisibility (gcd) 2

    Show that: b\neq 0 and a=bx+cy \Rightarrow (b,c)\mid (a,b)
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  2. #2
    o_O
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    Let d_1 = (a,b). This implies d_1 = {\color{red}a}m + bn for some m,n \in \mathbb{Z}

    We're given that: {\color{red}a} = bx+ cy

    So: d_1 = ({\color{red}bx + cy})m + bn \ \Leftrightarrow \ d_1 = {\color{magenta} b}(xm+n) + {\color{magenta}c}ym

    Can you see why (b,c) \mid d_1 ?
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  3. #3
    Sea
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    Yes... ... I can see...


    (b,c)=d \Rightarrow d|b and d|c \Rightarrow d|b(xm+n) and d|cym<br />
\Rightarrow d|b(xm+n)+cym \Rightarrow d| d_1





    Thanks a million...
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