Let $a,b \in \Bbb N $ with $\gcd(a,b)=1$. The equation $ax + by = ab$ has the obvious solution $(b, 0)$ in integers. Show, however, that it has no solution in positive integers.
Solutions of the equation $ax+by=ab$
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number-theory
elementary-number-theory
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0What happens when looking modulo $b$? And $a$? – 2012-10-18
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1Does this mean $by=moda$ and $ax=modb$? – 2012-10-18
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0Yes, $by = 0 \pmod a$ and $ax = 0 \pmod b$. Now you can use what you know about $\gcd(a,b)$. – 2012-10-18
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0So because $\gcd(a,b)=1$, $y$ must divide $a$ and $x$ must divide $b$? – 2012-10-18