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.