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Euclidean Algorithm

Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the GCD(greatest common divisor)

Implementation

int gcd(int a, int b) {
    if(!a) return b;
    return gcd(b % a, a);
}

Proof

This proof consists of two parts.
- part 1: proves that Euclidean Algorithm gives us a common factor of two integers.
- part 2: proves that the common divisor that Euclidean Algorithm produces is the largest possible

Part1

Euclidean Algorithm gives us a common factor of two integers($a, b$).

$$ \begin{aligned} a & = q_1b + r_1, &\text{where}(0<r<b)& \\ b & = q_2r_1 + r_2, &\text{where}(0<r_2<r_1)& \\ r_1 & = q_3r_2 + r_3, &\text{where}(0<r_3<r_2)&\\ &\vdots \\ r_i & = q_{i+2}r_{i+1} + r_{i+2}, & \text{where}( 0 < r_{i+2} < r_{i+1})& \\ &\vdots \\ r_{k-2} & = q_{k}r_{k-1} + r_{k}\\ r_{k-1} & = q_{k+1}r_k \\ \end{aligned} $$ From the last euqation, we know that $r_k|r_{k-1}$. So, we know that we can express $r_{k-1} = cr_k$. where $c$ is an integer. Now consider the previous equation. $$ \begin{aligned} r_{k-2} & = q_{k} r_{k-1} + r_{k} \\ & = q_{k} cr_{k} + r_{k} \\ & = r_{k} (q_{k}c + 1) \\ \end{aligned} $$ Thus, we have that $r_{k} | r_{k-2}$. In our equation previous to that one, we have $$ r_{k-3} = q_{k-1} r_{k-2} + r_{k-1} $$ From here, since $r_{k} | r_{k-1}$ and $r_{k}| r{k-2}$, using our rules of divisibility we have that $r_{k} | r_{k-3}$. As you can see, we can continue this process, considering each previous equation until we get to the last two, where we will find that $r_{k} | a$ and $r_{k} | b$. Thus, we find that Euclids algorithm gives us a common factor of a and b.

Part2

The common divisor Euclidean Algorithm produces is the largest possible.

We will start by assuming that $a$ and $b$ have a common factor $d$, and then show that $d | r_{k}$.

consider an arbitrary common factor d of $a$ and $b$. If $d$ is a common factor, we can rewrite $a$ and $b$ as follows:

$$ a = d a^{\prime}, b = d b^{\prime}, \text{where } d, a, b \text{ are all positive integers } $$

Now, consider the first euqation from Euclidean algorithm:

$$ \begin{aligned} a & = q_{1} b + r_{1} \\ r_{1} & = a - q_{1}b \\ & = da^{\prime} - q_{1} d b^{\prime} \\ & = d(a^{\prime} - q_{1}b^{\prime}) \end{aligned} $$

Thus, we have that $d|r_{1}$. Now, consider the second equation, and repeat the steps we did on the first, this time solving for $r_{2}$

$$ \begin{aligned} b & = q_{2}r_{1} + r_{2} \\ r_{2} & = b - q_{2}r_{1} \\ & = d b^{\prime} - q_{2}dr_1^{\prime} \\ & = d (b^{\prime} - q_{2} r_1^{\prime}) \end{aligned} $$

As you can see, we can continue this process through each of the equations until we hit the second to last one, where we will have

$$ \begin{aligned} r_{k-2} & = q_{k}r_{k-1} + r_{k} \\ r_{k} & = q_{k}r_{k-1} - r_{k-2} \\ & = q_{k}dr_{k-1}^{\prime} - d r_{k-2}^{\prime} \\ & = d(q_{k}r_{k-1}^{\prime} - r_{k-2}^{\prime}) \end{aligned} $$ Thus, $d| r_k$

But this says that any arbitrary common factor of $a$ and $b$ that we originally picked divides into $r_{k}$, the value that Euclidean algorithm produced. Since we know that $r_{k}$ is a common factor to both $a$ and $b$, this shows that is must be the largest possible common factor, or $gcd(a, b)$ $\blacksquare$

Extended Euclidean Algorithm

Given integers $a$ and $b$, there is always an integral solution to the equation $$ ax + by = gcd(a, b) $$ and we can find the values of $x$ and $y$.

implementation

not yet

Proof

Consider writing down the steps of Euclidean Algorithm

$$ \begin{aligned} a & = q_1b + r_1, &\text{where}(0<r<b)& \\ b & = q_2r_1 + r_2, &\text{where}(0<r_2<r_1)& \\ r_1 & = q_3r_2 + r_3, &\text{where}(0<r_3<r_2)&\\ &\vdots \\ r_i & = q_{i+2}r_{i+1} + r_{i+2}, & \text{where}( 0 < r_{i+2} < r_{i+1})& \\ &\vdots \\ r_{k-2} & = q_{k}r_{k-1} + r_{k} & \text{where}(0 < r_k < r_{k-1}) \\ r_{k-1} & = q_{k+1}r_k \\ \end{aligned} $$

Consider solving the second to last euqation for $r_k$. You get

$$ \begin{aligned} r_{k-2} & = q_{k}r_{k-1} + r_{k} \\ r_{k} & = r_{k-2} - q_{k}r_{k-1} \\ gcd(a, b) & = r_{k-2} - q_{k}r_{k-1} \end{aligned} $$

Now, solve the previous equation for $r_{k-1}$

$$ \begin{aligned} r_{k-3} & = q_{k-1}r_{k-2} + r_{k-1} \\ r_{k-1} & = r_{k-3} - q_{k-1}r_{k-2} \\ \end{aligned} $$

and we substitute this value in to the previous derived equation

$$ \begin{aligned} gcd(a, b) & = r_{k-2} - q_{k}r_{k-1} \\ & = r_{k-2} - q_k(r_{k-3} - q_{k-1}r_{k-2}) \\ & = r_{k-2}(1 - q_{k-1}) - q_kr_{k-3} \end{aligned} $$

Notice that now we have expressed $gcd(a, b)$ as a linear combination of $r_{k-2}$ and $r_{k-3}$. Next we can substitute for of $r_{k-2}$ in terms of $r_{k-3}$ and $r_{k-4}$, so that the $gcd(a, b)$ can be expressed as the linear combination of $r_{k-3}$ and $r_{k-4}$. Eventually, by continuing this process, $gcd(a, b)$ will be expressed as a linear combination of $a$ and $b$ as desired.

This process will be much easier to see with examples:

Find integers $x$ and $y$ such that

$$ 135x + 50y = 5 $$

Use Euclidean Algorithm to compute $gcd(135, 50)$

$$ \begin{aligned} 135 & = 2 \sdot 50 + 35 \cdots\text{\textcircled 1}\\ 50 & = 1 \sdot 35 + 15 \cdots\text{\textcircled 2}\\ 35 & = 2 \sdot 15 + 5 \cdots\text{\textcircled 3}\\ 15 & = 3 \sdot 5 \end{aligned} $$

Now, let's use the Extended Euclidean algorithm to solve the problem

$ 5 = 35 - 2 \sdot 15 $ from equation 3

But, we have that

$ 15 = 50 - 35 $ from euqation $\text{\textcircled 2}$

Now we, substitute this value into the previously derived equation: $$ \begin{aligned} 5 & = 35 - 2 \sdot 15 \\ 5 & = 35 - 2 \sdot (50 - 35) \\ 5 & = 3 \sdot 35 - 2 \sdot 50 \end{aligned} $$

Now, finally use the first equation to determine an expression for $35$ as a linear combination of $135$ and $50$ $$ 35 = 135 - 2 \sdot 50 \text{ from equation}\text{\textcircled 1} $$

Plug this into our last euqation:

$$ \begin{aligned} 5 & = 3 \sdot 35 - 2 \sdot 50 \\ 5 & = 3 \sdot (135 - 2 \sdot 50) - 2 \sdot 50 \\ 5 & = 3 \sdot 135 - 8 \sdot 50 \end{aligned} $$

So, a set of solutions to the equation is $x=3, y = -8$

This article is from 'COT3100Euclid01'

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