Euclid's Lemma for Prime Divisors/General Result/Proof 1

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Let $p$ be a prime number.

Let $\displaystyle n = \prod_{i \mathop = 1}^r a_i$.

Then if $p$ divides $n$, it follows that $p$ divides $a_i$ for some $i$ such that $1 \le i \le r$.

That is:

$p \divides a_1 a_2 \ldots a_n \implies p \divides a_1 \lor p \divides a_2 \lor \cdots \lor p \divides a_n$


As for Euclid's Lemma for Prime Divisors, this can be verified by direct application of general version of Euclid's Lemma for irreducible elements.


Source of Name

This entry was named for Euclid.