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

Lemma

Let $p$ be a prime number.

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

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

That is:

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

Proof

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.

$\blacksquare$

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Source of Name

This entry was named for Euclid.