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Math Help - Prime Numbers

  1. #1
    Junior Member
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    Prime Numbers

    Here is my problem:

    Let a,b be nonzero integers. Prove that (a,b) = 1 if and only if (a+b, ab) = 1.

    Any suggestion on where to start?
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  2. #2
    Senior Member roninpro's Avatar
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    You could try playing around with Bezout's identity.

    (From right to left) Suppose that \gcd(a+b,ab)=1. There exist integers x,y such that (a+b)x+(ab)y=1. Distributing the terms, ax+bx+aby=1, and a(x+by)+bx=1. This shows that \gcd(a,b) must divide 1. Therefore, \gcd(a,b)=1.

    You can try a similar argument for the other direction. (You may need to insert some terms into the equation by adding and subtracting the same thing.)
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  3. #3
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    Alternatively:

    <br />
\gcd(a+b,ab) = 1 \Leftrightarrow \gcd(a+b,a) = \gcd(a+b,b) = 1 \Leftrightarrow \gcd(a,b)=1<br />
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