This is maybe a stupid question but I have to ask,
Is it correct that for all $d\in \mathbb N$ exists two numbers $a,b\in\mathbb Z$ such that $\text{gcd}(a,b)=d$?
I think that the answer is NO e.g let's take $d=0$
Am I correct?
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Error 404
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Is $0$ a natural number? – Igor Rivin Mar 05 '16 at 20:57
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Yes it is, why not? – Error 404 Mar 05 '16 at 20:59
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@Error404 : Ok, Is it true for non zero natural numbers? – Mar 05 '16 at 21:00
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For non zero I think that the answer will be Yes – Error 404 Mar 05 '16 at 21:01
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how? @Error404.... – Mar 05 '16 at 21:01
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1Some authors will assume $\mathbb{N}$ not to include $0$, so it's best to make this explicit. – Fryie Mar 05 '16 at 21:08
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You are correct if $0 \in \Bbb N$ For all other numbers, there are.
Ross Millikan
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Depending on how we define $\gcd$. Some definitions make $\gcd(0,0)=0$. – Wojowu Mar 05 '16 at 21:33
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And with good reason, see http://math.stackexchange.com/questions/495119/what-is-gcd0-0. – Fryie Mar 05 '16 at 21:45