Concise proof that every common divisor divides GCD without Bezout's identity?

Question

In the integers, it follows almost immediately from the division theorem and the fact that $a \mid x,y \implies a \mid ux + vy$ for any $u, v \in \mathbb{Z}$ that the least common multiple of $a$ and $b$ divides any other common multiple.

In contrast, proving $e\mid a,b \implies e\mid\gcd(a,b)$ seems to be more difficult. In Elementary Number Theory by Jones & Jones, they do not try to prove this fact until establishing Bezout's identity. This Wikipedia page has a proof without Bezout's identity, but it is convoluted to my eyes.

I tried my hand at it, and what I got seems no cleaner:

Proposition: If $e \mid a,b$, then $e \mid \gcd(a,b)$.

Proof: Let $d = \gcd(a,b)$. Then if $e \nmid d$, by the division theorem there's some $q$ and $c$ such that $d = qe + c$ with $0 < c < r$.

We have $a = k_1 d$ and $b = k_2 d$, so by substituting we obtain $a = k_1 (qe + c)$ and $b = k_2 (qe + c)$. Since $e$ divides both $a$ and $b$, it must divide both $k_1 c$ and $k_2 c$ as well. This implies that both $k_1 c$ and $k_2 c$ are common multiples of $c$ and $r$.

Now let $l = \operatorname{lcm}(r, c)$. $l$ divides both $k_1 c$ and $k_2 c$. Since $l = \phi c$ for some $\phi$, we have $\phi | k_1, k_2$, so $d \phi | a, b$.

But we must have $\phi > 1$ otherwise $l = c$, implying $r \mid c$, which could not be the case since $c < r$. So $d \phi$ is a common divisor greater than $d$, which is a contradiction. $\Box$

Question: Is there a cleaner proof I'm missing, or is this seemingly elementary proposition just not very easy to prove without using Bezout's identity?

Answer 1

One easy and insightful way is to use the proof below. It essentially constructs $\rm\,gcd\,$ from $\rm\,lcm\,$ by employing duality between minimal and maximal elements - see the Remark below. This is essentially how the linked Wikipedia proof works, but there the innate duality is obfuscated by the presentation. Below is a proof structured so that this fundamental duality is brought to the fore.

$\rm{\bf Theorem}\quad c\mid a,b\iff c\mid d,\ \ $ for $\rm\ \ d = ab/lcm(a,b).\ $ $\rm\color{#0a0}{Hence}$ $\rm\ d = gcd(a,b)$

$\rm{\bf Proof}\qquad\ \ \, c\mid a,b \iff a,b\mid ab/c \iff lcm(a,b)\mid ab/c \iff c\mid ab/lcm(a,b)$

$\rm\color{#0a0}{Generally}\,$ if $\rm\, c\mid a,b\iff c\mid d\ $ then $\rm\ d = \gcd(a,b)\ $ up to unit factors, i.e. they're associate (this is the definition of a gcd $\rm\,d\,$ of $\rm\,a,b\,$ in an integral domain).

Indeed setting $\rm\:c = d\:$ in direction $(\Leftarrow)$ shows that $\rm\:d\mid a,b,\:$ i.e. $\rm\:d\:$ is a common divisor of $\rm\:a,b.\:$ Conversely, by direction $(\Rightarrow)$ we deduce that $\rm\:d\:$ is divisible by every common divisor $\rm\:c\:$ of $\rm\:a,b,\:$ thus $\rm\:c\mid d\:\Rightarrow\: c\le d,\:$ so $\rm\:d\:$ is a greatest common divisor (both divisibility and magnitude-wise).

Remark $\ $ The proof shows that, in any domain, if $\rm\:lcm(a,b)\:$ exists then $\rm\:gcd(a,b)\:$ exists and $\rm\ gcd(a,b)\,lcm(a,b) = ab\ $ up to unit factors, i.e. they are associate. The innate duality in the proof is clarified by employing the involution $\rm\ x'\! = ab/x\ $ on the divisors of $\rm\:ab.\:$ Let's rewrite the proof using this involution (reflection), as in this answer (and here).

Notice that $\rm\ x\,\mid\, y\:\color{#c00}\iff\: y'\mid x'\,\ $ by $\smash[t]{\,\ \rm\dfrac{y}x = \dfrac{x'}{y'} \ }$ by $\rm\, \ yy' = ab = xx',\ $ so rewriting using this

$\begin{eqnarray}\rm the\ proof\ \ \ c\mid a,b &\iff&\rm b,\,a\mid ab/c &\iff&\rm lcm(b,\,a)\mid ab/c &\iff&\rm c\mid ab/lcm(b,a)\\[.5em] \rm becomes\ \ \ \ c\mid a,b &\color{#c00}\iff&\rm a',b'\mid c' &\iff&\rm lcm(a',b')\mid c' &\color{#c00}\iff&\rm c\mid lcm(a',b')'\end{eqnarray}$

Now the innate duality is clear: $\rm\ gcd(a,b)\,=\,lcm(a',b')'\ $ by the $\rm\color{#0a0}{above}$ gcd characterization.

Answer 2

We can gain some insight by seeing what happens for other rings. A GCD domain is an integral domain $D$ such that $\gcd$s exist in the sense that for any $a, b \in D$ there exists an element $\gcd(a, b) \in D$ such that $e | a, e | b \Rightarrow e | \gcd(a, b)$. A Bézout domain is an integral domain satisfying Bézout's identity.

Unsurprisingly, Bézout domains are GCD domains, and the proof is the one you already know. It turns out that the converse is false, so there exist GCD domains which are not Bézout domains; Wikipedia gives a construction.

(But if you're allowing yourself the division algorithm, why the fuss? The path from the division algorithm to Bézout's identity is straightforward. In all of these proofs for $\mathbb{Z}$ the division algorithm is doing most of the work.)

Answer 3

Proof using linear diophantine equations; Note: if $d=\gcd(a,b)$, then there exist integers $x$ and $y$ such that $$d=ax+by$$

Suppose $c\mid a$ and $c\mid b$ and $d=\gcd(a,b)$. Then, by definition of divisibility, $a=cl$ & $b=ck$, for some integers $l$ & $k$. By definition of gcd, $d=ax+by$ for some integers $x$ & $y$. By substitution, $$d=(cl)x + (ck)y =clx + cky =c(lx+ky) =cm$$ by closure, with $m=lx+ky$. Therefore $$d=cm$$ But by definition of divisibility, this implies $c\mid d$ and $m\mid d$ Therefore $c\mid d$.

QED :)

Answer 4

You might also use the fact that each natural number can be uniquely represented as a product of prime powers to prove this fact without Bezout's Theorem.

This is called Fundamental Theorem of Arithmetic

For example d=p1^d1*..*pn^dn where p1 to pn are primes. The numbers p1 to pn are called then prime factors of d

Idea of the proof argument: Since gcd(a,b) divides a and b each prime factor of gcd(a,b) has to be a common prime factor of a and b.

For the same reason each prime factor of d has to be a common prime factor of a and b.

And since gcd(a,b) is the greatest common divisor of a,b (by definition), each prime factor of d has to be then a prime factor of gcd(a,b), for if there would be at least one prime factor p of d which isn't a prime factor of gcd(a,b), then the product gcd(a,b)*p would be again a common divisor of both a and b, greater than gcd(a,b), which would be a contradiction.

Answer 5

Just think of the Euclidean algorithm for computing the GCD of two numbers. Let $a, b$ with say $a<b$ be two integers. Then the GCD of $a$ and $b$ is also the GCD of $a$ and $b-a$, and now we can iterate. For intance:

1. The GCD of $28$ and $8$ is the same as that of $20$ and $8$.
2. Which is the same as that of $12$ and $8$.
3. Which is the same as that of $4$ and $8$.
4. Which is the same as that of $4$ and $4$.
5. Which is obviously $4$.

Eventually, we must come down to calculating the GCD of a number with itself, since at each stage one of the numbers in our pair stays the same, and the other gets smaller. This has to stop eventually, but as long as the two numbers are distinct we can always subtract the smaller from the larger, so the only way it can stop is if they end up being equal, at which point that number will be the GCD of the original pair.

Now you just have to realize that this same logic applies to any divisor of the two numbers, not jut the greatest one. I could have said "any divisor of $28$ and $8$ is a divisor of $20$ and $8$..." and eventually had "...is a divisor of $4$". Since the number we arrive at at the end is the GCD, any divisor of the two original numbers divides the GCD.