Existence of Greatest Common Divisor/Proof 2
Theorem
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.
Proof
By definition of greatest common divisor, we aim to show that there exists $c \in \Z_{>0}$ such that:
\(\ds c\) | \(\divides\) | \(\ds a\) | ||||||||||||
\(\ds c\) | \(\divides\) | \(\ds b\) |
and:
- $d \divides a, d \divides b \implies d \divides c$
Consider the set $S$:
- $S = \set {s \in \Z_0: \exists x, y \in \Z: s = a x + b y}$
$S$ is not empty, because by setting $x = 1$ and $y = 0$ we have at least that $a \in S$.
From the Well-Ordering Principle, there exists a smallest $c \in S$.
So, by definition, we have $c > 0$ is the smallest such that $c = a x + b y$ for some $x, y \in \Z$.
Let $d$ be such that $d \divides a$ and $d \divides b$.
Then from Common Divisor Divides Integer Combination:
- $d \divides a x + b y$
That is:
- $d \divides c$
We have that:
\(\ds \exists t, u \in \Z: \, \) | \(\ds a\) | \(=\) | \(\ds c t + u:\) | \(\ds 0 \le u < c\) | Division Theorem | |||||||||
\(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds a x t + b y t + u\) | Definition of $c$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds u\) | \(=\) | \(\ds r \paren {1 - x t} + b \paren {-y t}\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds u\) | \(=\) | \(\ds 0\) | as $u < c$ and the definition of $c$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds c\) | \(\divides\) | \(\ds a\) | Definition of Divisor of Integer |
- $c \divides b$
Now suppose $c'$ is such that:
\(\ds c'\) | \(\divides\) | \(\ds a\) | ||||||||||||
\(\ds c'\) | \(\divides\) | \(\ds b\) |
and:
- $d \divides a, d \divides b \implies d \divides c'$
Then we have immediately that:
- $c' \divides c$
and by the same coin: $c \divides c'$
and so:
- $c = c'$
demonstrating that $c$ is unique.
$\blacksquare$
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 6$: The division process in $I$
- 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: Theorem $1.2$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $3$: The Integers: $\S 11$. Highest Common Factor: Theorem $19$