Existence of Greatest Common Divisor

From ProofWiki
Jump to navigation Jump to search

Theorem

$\forall a, b \in \Z: a \ne 0 \lor b \ne 0$, there exists a largest $d \in \Z_{>0}$ such that $d \divides a$ and $d \divides b$.

The greatest common divisor of $a$ and $b$ always exists.


Proof

Proof of Existence

We have that:

$\forall a, b \in \Z: 1 \divides a \land 1 \divides b$

so $1$ is always a common divisor of any two integers.

$\Box$


Proof of there being a Largest

As the definition of $\gcd$ shows that it is symmetric, we can assume without loss of generality that $a \ne 0$.

First we note that from Absolute Value of Integer is not less than Divisors:

$\forall c \in \Z: \forall a \in \Z_{>0}: c \divides a \implies c \le \size c \le \size a$

The same applies for $c \divides b$.


Now we have three different results depending on $a$ and $b$:

\(\displaystyle a \ne 0 \land b \ne 0\) \(\implies\) \(\displaystyle \gcd \set {a, b} \le \min \set {\size a, \size b}\)
\(\displaystyle a = 0 \lor b = 0\) \(\implies\) \(\displaystyle \gcd \set {a, b} = \max \set {\size a, \size b}\)
\(\displaystyle a = b = 0\) \(\implies\) \(\displaystyle \forall x \in \Z: x \divides a \land x \divides b\)


So if $a$ and $b$ are both zero, then any $n \in \Z$ divides both, and there is no greatest common divisor.

This is why the proviso that $a \ne 0 \lor b \ne 0$.


So we have proved that common divisors exist and are bounded above.

Therefore, from Set of Integers Bounded Above by Integer has Greatest Element there is always a greatest common divisor.

$\blacksquare$


Sources