Proposition: Greatest Common Divisor of More Than Two Numbers

Let $n_1,n_1,\ldots,n_k\in\mathbb Z$ be integers. The simultanous greatest common divisor of all these numbers can be calculated by the recursive formula


This calculation is independent of the order, in which the recursive formula is used, following the associativity and commutativity of the $\gcd$. In particular, for any integers $a,b,c\in\mathbb Z$ we have $$\gcd(\gcd(a,b),c))=\gcd(a,\gcd(b,c))$$ and $$\gcd(a,b)=\gcd(b,a).$$

Proofs: 1

Proofs: 1
Propositions: 2 3

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  1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
  2. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927