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State/Greatest common divisors can be computed pairwise
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Proposition: (Greatest common divisors can be computed pairwise) Suppose that $k \in \NN$, and $k > 2$. Suppose that $a_1, \ldots, a_k$ are integers, and that $a_1 \neq 0$. Then, $$GCD(a_1, \ldots, a_k) = GCD(GCD(a_1, a_2), a_3, \ldots, a_k),$$ where the GCD is taken to mean the greatest common positive divisor.
Logical Connections
This statement logically relies on the following definitions and statements: Def/Greatest common divisor, State/Mutual divisibility of natural numbers implies equality
The following statements and definitions rely on the material of this page: State/Linear Diophantine equations can be solved with the Euclidean algorithm
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