Exercise 1.2.19 (integers relatively prime in pairs are relatively prime)

Question

Prove that any set of integers that are relatively prime in pairs are relatively prime.

Richard Ganaye · Accepted Answer

Since the text doesn't define the gcd of an infinite set of integers, I suppose that this set is finite.

\begin{proof}
Let $A = \{a_1,a_2,\ldots,a_n\}$ a finite set of integers with $n>1$. Since $a_1,a_2,\ldots,a_n$ are relatively prime in pairs, $a_1 \wedge a_2 = 1$.
By definition, $a_1 \wedge a_2 \wedge \cdots\wedge a_n \mid a_1$ and $a_1 \wedge a_2 \wedge \cdots\wedge a_n  \mid a_2$, therefore
$$a_1 \wedge a_2 \wedge \cdots\wedge a_n \mid a_1 \wedge a_2,$$
where $a_1 \wedge a_2 \wedge \cdots\wedge a_n \geq 0$, we obtain $a_1 \wedge a_2 \wedge \cdots\wedge a_n = 1$, so $a_1,a_2,\ldots,a_n$ are relatively prime.
 \end{proof}

Exercise 1.2.19 (integers relatively prime in pairs are relatively prime)

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Exercise 1.2.19 (integers relatively prime in pairs are relatively prime)

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