Euclid's Lemma

Theorem

Let $a, b, c \in \Z$.

Let $a \divides b c$, where $\divides$ denotes divisibility.

Let $a \perp b$, where $\perp$ denotes relative primeness.

Then $a \divides c$.

Proof 1

Follows directly from Integers are Euclidean Domain.

$\blacksquare$

Proof 2

Let $a, b, c \in \Z$.

We have that $a \perp b$.

That is:

$\gcd \set {a, b} = 1$

where $\gcd$ denotes greatest common divisor.

From Bézout's Identity, we may write:

$a x + b y = 1$

for some $x, y \in \Z$.

Upon multiplication by $c$, we see that:

$c = c \paren {a x + b y} = c a x + c b y$

Now note that $c a x + c b y$ is an integer combination of $a c$ and $b c$.

So, since:

$a \divides a c$

and:

$a \divides b c$

it follows from Common Divisor Divides Integer Combination that:

$a \divides \paren {c a x + c b y}$

However:

$c a x + c b y = c \paren {a x + b y} = c \cdot 1 = c$

Therefore:

$a \divides c$

$\blacksquare$

Source of Name

This entry was named for Euclid.

Also see

• Euclid's Lemma for Prime Divisors

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Some Properties of $\Z$: Exercise $2.5$