Euclidean Algorithm

Algorithm
The Euclidean Algorithm (or Euclid's Algorithm/Euclidean Division Algorithm) is a method for finding the greatest common divisor (GCD) of two integers $$a$$ and $$b$$.

Let $$a, b \in \Z$$ and $$a \ne 0 \lor b \ne 0$$.

The steps are:
 * 1) Start with $$(a,b)$$ such that $$\left|{a}\right| \ge \left|{b}\right|$$. If $$b = 0$$ then you are done and the GCD is $$a$$.
 * 2) If $$b \neq 0$$ then you take the principal remainder $$r$$ of $$\frac{a}{b}$$.
 * 3) Set $$a \gets b, b \gets r$$ (and thus $$\left|{a}\right| \ge \left|{b}\right|$$ again).
 * 4) Repeat these steps until $$b=0$$.

Thus the GCD of $$a$$ and $$b$$ is the value of the variable $$a$$ at the end of the algorithm.

It can be seen from the definition that the Euclidean algorithm is indeed an algorithm in the strict mathematical sense.

Proof
Suppose $$a, b \in \Z$$ and $$a \lor b \ne 0$$.

From the Division Theorem, $$a = q b + r$$ where $$0 \le r < \left|{b}\right|$$.

From GCD with Remainder, the GCD of $$a$$ and $$b$$ is also the GCD of $$b$$ and $$r$$.

Therefore, we may search instead for $$\gcd \left\{{b, r}\right\}$$.

Since $$|r| < |b|$$ and $$b \in \Z$$, we will reach $$r=0$$ after finitely many steps.

At this point, $$\gcd \left\{{r, 0}\right\} = r$$ from GCD with Zero.

Demonstration
We can investigate in more detail what happens when we apply the Division Theorem repeatedly to $$a$$ and $$b$$.

$$ $$ $$ $$ $$ $$

From the Division Theorem, we know that the remainder is always strictly less than the divisor.

That is, in $$a = q b + r, 0 \le r < \left|{b}\right|$$.

So we know that:

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

So the algorithm has to terminate.

Algorithmic Nature

 * Finiteness: As has been seen, the algorithm always terminates after a finite number of steps.
 * Definiteness: Each of the steps is precisely defined.
 * The inputs are $$a$$ and $$b$$.
 * The output is $$\gcd \left\{{a, b}\right\}$$.
 * Effectiveness: Each operation is finite in extent and can be effectively performed.

Constructing an Integer Combination
Having done this, we are now in a position to find a solution to $$\gcd \left\{{a, b}\right\} = x a + y b$$ for $$x$$ and $$y$$ by Bézout's Identity it is possible to find such an $$x,y \in \Z$$.

Working back through the equations above, we get:

$$ $$ $$ $$ $$

... and so on. The algebra gets messier the further up the tree you go, and is not immediately enlightening.

(However, see Rational Numbers and SFCFs are Equivalent‎ for an application of the Euclidean algorithm in a slightly different context.)

Thus eventually we arrive at $$\gcd \left\{{a, b}\right\} = x a + y b$$ where $$x$$ and $$y$$ are numbers made up from some algebraic cocktail of the coefficients of the terms involving the remainders from the various applications of the division theorem.

Note that $$a \backslash b \implies \gcd \left\{{a, b}\right\} = a$$, from GCD of Integer and Divisor.