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I‘m supposed to prove the following without using prime factorisation:

Let a1 $\neq$ 0, a2, … , an $\in \mathbb{Z}$ be pairwise co-primes. Show that a1 and a2 * … * an are also coprimes.

It’s so simple, but how can we do that without prime fact.? In class we just had Bezouts Lemma, the Chinese remainder theorem and Fermats little theorem resp. Eulers theorem.

Bill Dubuque
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    Well, if you have Bezout then presumably you have the fact that if a prime $p$ divides a product $mn$ then $p$ divides at least one of $m,n$. – lulu May 07 '22 at 18:39
  • Use Bezout theorem to translate the assumption and to prove that $a_1$ and $a_2a_3$ are coprime. – Christophe Leuridan May 07 '22 at 19:48

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