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Assume that $$ \gcd(a,b)=ax+by $$ for some $a, b, x, y \in \mathbb Z$. How do I show that $\gcd(x,y)=1$? (Hint: contradiction.)

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What do you mean with "syt"? –  ulead86 Sep 5 '11 at 10:46
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You have asked many questions and accepted few answers. Yet you ask another question. This does not compute. –  Gerry Myerson Sep 5 '11 at 10:52
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Short answer: you can't. Or at least, just because the gcd can be expressed in that form doesn't mean that it is 1. –  Josh Chen Sep 5 '11 at 10:55
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@Josh: I think you misread the question. You are asked to show that $\gcd(x,y)=1$, not $\gcd(a,b)=1$. (Perhaps you are seeing the same formatting error that I see -- the LaTex expression $\gcd(x,y)=1$ is above the first line instead of below it.) –  TonyK Sep 5 '11 at 11:37
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@alvoutila: I am not sure by the way the question is put whether you know the answer to the question and are giving me a hint, or whether the question comes with a hint, and you have not been able to solve it. In case it helps, the hint suggests assuming that there is a number c > 1 which divides both $x$ and $y$. –  Mark Bennet Sep 5 '11 at 13:20
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3 Answers

Suppose $\gcd(a,b) = ax + by = d$. Then $\exists \ u, v \in \mathbb{N}$ such that $a = u \cdot d$ and $b = v \cdot d$. So then $d = ax + by = d (u x + v y)$, or in other words $u x + v y = 1$ with $u,v \in \mathbb{N}$. So $\gcd(x,y) = 1$.

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HINT $\rm\ \ \ (a,b)\:|\:a,b,\ \ c\:|\:x,y\ \Rightarrow\ (a,b)\:c\ |\ a\ x + b\ y\: =\: (a,b)\ \Rightarrow\ c\:|\: 1\:.\:$ Put $\rm\ c = (x,y)\:.$

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When I saw this question in the feed, I thought: "Shouldn't Bill be the one giving hints ?!?!"$$$$ :) –  The Chaz 2.0 Sep 5 '11 at 14:14
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@The Only when SE has neural RSS feed I can be FGITW while asleep! –  Bill Dubuque Sep 5 '11 at 14:24
    
I'm sure Skynet is working on it! –  The Chaz 2.0 Sep 5 '11 at 14:57
    
@Thi I've slightly altered the presentation. If this doesn't answer your query please let me know. –  Bill Dubuque Sep 5 '11 at 15:15
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A proof by contradiction: Suppose $\gcd(x,y) = d > 1$. Then $\exists \ u,v \in \mathbb{Z}$ such that $x = u \cdot d$ and $y = v \cdot d$. So then $\gcd(a,b) = ax + by = d(au + bv)$, so $au + bv = \gcd(a,b)/d < \gcd(a,b)$ with $u, v \in \mathbb{Z}$. But this contradicts the fact that $\gcd(a,b)$ is the greatest common divisor of $a$ and $b$. Therefore our assumption $d > 1$ was wrong, and $\gcd(x,y) = 1$.

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I don't get the point. Why do you get that gcd(a,b) isn't the greatest common divisor? How gcd(a,b)/d<gcd(a,b) indicates that gcd(a,b) isn't the greatest common divisor? –  laovultai Sep 5 '11 at 15:41
    
In other words, you just use fact that gcd(a,b) is as follows( definition) to conclude contradiction with the assumption(antithesis)? –  laovultai Sep 5 '11 at 16:33
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