Euclid's Lemma for Prime Divisors/Proof 1

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Lemma

Let $p$ be a prime number.

Let $a$ and $b$ be integers such that:

$p \divides a b$

where $\divides$ means is a divisor of.

Then $p \divides a$ or $p \divides b$.


Proof

We have that the integers form a Euclidean domain.

Then from Irreducible Elements of Ring of Integers we have that the irreducible elements of $\Z$ are the primes and their negatives.


The result then follows directly from Euclid's Lemma for Irreducible Elements.

$\blacksquare$


Source of Name

This entry was named for Euclid.