Existence of Greatest Common Divisor/Proof 4

From ProofWiki
Jump to navigation Jump to search


Let $a, b \in \Z$ be integers such that $a \ne 0$ or $b \ne 0$.

Then the greatest common divisor of $a$ and $b$ exists.


From the Euclidean Algorithm, we have calculated a sequence $\tuple {r_1, r_2, \ldots r_{n - 2}, r_{n - 1}, r_n}$ such that:

$b > r_1 > r_2 > \dotsb > r_{n - 2} > r_{n - 1} > r_n = 0$

We have that:

$r_{n - 1} \divides a$


$r_{n - 1} \divides b$

Working backwards from the final equation, we see that:

$r_k \divides r{k - 1}$

for all $k$ such that $1 < k \le n$.

Hence, if $d \divides a$ and $d \divides b$, we can use induction to proceed through the Euclidean Algorithm and see that $d$ divides $r_1, r_2, \ldots, r_{n - 2}, r_{n - 1}$.

Thus we see that $r_{n - 1}$ fulfils the criteria to be the greatest common divisor of $a$ and $b$.


Suppose $c_1$ and $c_2$ are both greatest common divisors of $a$ and $b$.

Then by definition there exist $g, h \in \Z_{>0}$ such that:

$g c_1 = c_2$
$h c_2 = c_1$


$c_2 = g h c_2$

and so:

$g h = 1$

That is:

$g = h = 1$

and so:

$c_1 = c_2$

That is, the greatest common divisor of $a$ and $b$ is unique.