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Assume $a,b,c,d\in\Bbb N_{>1}$ and $a\neq c$.

Does having both $ab=cd$, $gcd(a,c)=1$, $a\geq d$ and $c\geq b$ imply that $a=d$ and $b=c$ holds?

Turbo
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A counterexample is $a=1$, $b=c=d=0$. However, there are no counterexamples if you require $b\neq 0$. Since $a$ and $c$ are relatively prime but $c$ divides $ab$, $c$ must divide $b$. Since $b\neq 0$ and $c\geq b$, this can only happen if $b=c$. It then follows that $a=d$.

Eric Wofsey
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  • don't think $gcd(a,c)$ is defined for $ac=0$ case. Also $0\not\in\Bbb N$. – Turbo Nov 13 '16 at 08:52
  • It's quite standard to say $\gcd(0,n)=n$ for any $n\in\mathbb{N}$. See the answers to http://math.stackexchange.com/questions/1386651/is-this-gcd0-0-0-a-wrong-belief-in-mathematics-or-it-is-true-by-conventi for why this makes sense. – Eric Wofsey Nov 13 '16 at 08:54
  • ok still $0\not\in\Bbb N$. – Turbo Nov 13 '16 at 08:58