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Given numbers $a,b,c\in\mathbb N$, say $k$ be the smallest positive linear combination of $\{a,b,c\}$. Now consider $a$, if:

$$ a=kq+r, 0\lt r\lt k, $$

then there exists a smaller positive linear combination of $\{a,b,c\}$, a contradiction, so $r=0$. The idea for $b,c$ is the same, we get $k$ divides $a,b,c$. Let $d=\gcd(a,b,c)$, since $d$ divides $k$ and $k\le d$, $k=d$.

So $\gcd(a,b,c)$ can be defined to be the smallest positive linear combination of $\{a,b,c\}$, but how to find the coefficient of $a,b,c$ respectively?

linear_combinatori_probabi
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1 Answers1

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From extended Euclidean algorithm, we can find $\alpha, \beta, \gamma, \delta$ such taht

$$gcd(a,b)=\alpha a + \beta b$$

$$gcd(a,b,c)=gcd(gcd(a,b), c)= \gamma c + \delta \gcd(a,b)$$

Hence

$$\gcd(a,b,c) = \alpha \delta a + \beta \delta b + \gamma c$$

Siong Thye Goh
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