Let $a,b \in \Bbb N$ with $\gcd(a,b)=1$. Show that for every integer $n>ab$ the equation $ax+by=n$ has a solution in positive integers $x,y$. (Take $(x,y)$ with $y \leq 0$ and $x$ minimal).
Number Theory Problem $ax+by=n$ for $n>ab$
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elementary-number-theory
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0I assume you have learned about [Bézout's lemma](http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity)? So there are solutions to the equation, perhaps not with positive $x$ and $y$. The solution is not unique. So you take the hint at the end of the problem statement and work with it. What happens now? – 2012-10-18