Euclidean algorithm

From AoPSWiki

The Euclidean algorithm (also known as the Euclidean division algorithm or Euclid's algorithm) is an algorithm that finds the greatest common divisor (GCD) of two elements of a Euclidean domain, the most common of which is the nonnegative integers $\mathbb{Z}{\geq 0}$ , without factoring them.

Main idea and Informal Description

The basic idea is to repeatedly use the fact that $\gcd({a,b}) \equiv \gcd({a,a - b})$

If we have two non-negative integers $a,b$ with $a\ge b$ and $b=0$ , then the greatest common divisor is ${a}$ . If $a\ge b>0$ , then the set of common divisors of ${a}$ and $b$ is the same as the set of common divisors of $b$ and $r$ where $r$ is the remainder of division of ${a}$ by $b$ . Indeed, we have $a=mb+r$ with some integer $m$ , so, if ${d}$ divides both ${a}$ and $b$ , it must divide both ${a}$ and $mb$ and, thereby, their difference $r$ . Similarly, if ${d}$ divides both $b$ and $r$ , it should divide ${a}$ as well. Thus, the greatest common divisors of ${a}$ and $b$ and of $b$ and $r$ coincide: $GCD(a,b)=GCD(b,r)$ . But the pair $(b,r)$ consists of smaller numbers than the pair $(a,b)$ ! So, we reduced our task to a simpler one. And we can do this reduction again and again until the smaller number becomes $0$

General Form

Start with any two elements $a$ and $b$ of a Euclidean Domain

If $b=0$ , then $\gcd(a,b)=a$ .
Otherwise take the remainder when ${a}$ is divided by $a (\bmod {b})$ , and find $\gcd(a,a \bmod {b})$ .
Repeat this until the remainder is 0.

$a (\bmod b) \equiv r_1$
$b (\bmod r_1) \equiv r_2$
$\vdots$
$r_{n-1} (\bmod r_n) \equiv 0$
Then $\gcd({a,b}) = r_n$

Usually the Euclidean algorithm is written down just as a chain of divisions with remainder:

for $r_{k+1} < r_k < r_{k-1}$
$a = b \cdot q_1+r_1$
$b = r_1 \cdot q_2 + r_2$
$r_1 = r_2 \cdot q_3 + r_3$
$\vdots$
$r_{n-1} = r_n \cdot q_{n+2} +0$
and so $\gcd({a,b}) = r_n$

Simple Example

To see how it works, just take an example. Say $a=112,b=42$ . We have $112\equiv 28\pmod {42}$ , so ${\gcd(112,42)}=\gcd(42,28)$ . Similarly, $42\equiv 14\pmod {28}$ , so $\gcd(42,28)=\gcd(28,14)$ . Then $28\equiv {0}\pmod {14}$ , so ${\gcd(28,14)}={\gcd(14,0)} = 14$ . Thus $\gcd(112,42)=14$ .

${112 = 2 \cdot 42 + 28 \qquad (1)}$
$42 = 1\cdot 28+14\qquad (2)$
$28 = 2\cdot 14+0\qquad (3)$

Linear Representation

An added bonus of the Euclidean algorithm is the "linear representation" of the greatest common divisor. This allows us to write $\gcd(a,b)=ax+by$ , where $x,y$ are some elements from the same Euclidean Domain as $a$ and $b$ that can be determined using the algorithm. We can work backwards from whichever step is the most convenient.