Euclid's Lemma for Prime Divisors

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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 $p \divides b$.


General Result

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:

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


Corollary

Let $p, p_1, p_2, \ldots, p_n$ be primes such that:

$p \divides \displaystyle \prod_{i \mathop = 1}^n p_i$

Then:

$\exists i \in \closedint 1 n: p = p_i$


Proof 1

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.

$\blacksquare$


Proof 2

Let $p \divides a b$.


Suppose $p \nmid a$.

Then from the definition of prime:

$p \perp a$

where $\perp$ indicates that $p$ and $a$ are coprime.

Thus from Euclid's Lemma it follows that:

$p \divides b$


Similarly, if $p \nmid b$ it follows that $p \divides a$.

So:

$p \divides a b \implies p \divides a$ or $p \divides b$

as we needed to show.

$\blacksquare$


Proof 3

Let $p \divides a b$.


Suppose $p \nmid a$.

Then by Proposition $29$ of Book $\text{VII} $: Prime not Divisor implies Coprime:

$p \perp a$

As $p \divides a b$, it follows by definition of divisor:

$\exists e \in \Z: e p = a b$

So by Proposition $19$ of Book $\text{VII} $: Relation of Ratios to Products‎:

$p : a = b : e$

But as $p \perp a$, by:

Proposition $21$ of Book $\text{VII} $: Coprime Numbers form Fraction in Lowest Terms

and:

Proposition $20$ of Book $\text{VII} $: Ratios of Fractions in Lowest Terms

it follows that:

$p \divides b$

Similarly, if $p \perp b$ then $p \divides a$.


Hence the result.

$\blacksquare$


Also presented as

Some sources present this as:

Let $p$ be a prime number.

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

$a b \equiv 0 \pmod p$

Then either $a \equiv 0 \pmod p$ or $b \equiv 0 \pmod p$.


Also known as

Some sources give the name of this as Euclid's first theorem.


Source of Name

This entry was named for Euclid.


Also see

Some sources use this property to define a prime number.


Sources