If two distinct primes p,q divides a number z, will their product $pq$ also divide z, that is $pq|z$? And how can I prove this. Thanks,
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2Isn't this a more or less immediate consequence of the unique factorization theorem (fundamental theorem of arithmetic)? – Brian Tung Oct 03 '20 at 19:46
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Use the Fundamental Theorem of arithmetic. – Dr. Mathva Oct 03 '20 at 19:46
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2If you don't want to go all the way to the fundamental theorem, https://en.wikipedia.org/wiki/Euclid's_lemma might help. – Arthur Oct 03 '20 at 19:46
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Euclids lemma did the job, thanks everyone. – ghetto_department Oct 03 '20 at 19:50
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$\mathrm{gcd}(a,b)=1$ means in your setup, that $a\nmid b$ and $b\nmid a$. – Fakemistake Oct 04 '20 at 07:24
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Notice that $p\mid z\implies z=p\cdot k$ for some integer $k$. At the same time, Eucklid's Lemma establishes that $$q\mid z\iff q\mid p\cdot k\implies q\mid p\;\text{ or/and }\;q\mid k$$ Observe that $q\mid p$ is nonsense, since both $p$ and $q$ are primes. Therefore $q\mid k\implies k=q\cdot m$ for some integer $m$. But then $z=p\cdot k=p\cdot q\cdot m\implies pq\mid z$.
Dr. Mathva
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