## Bezout’s identity

For non-zero integers $a$ and $b$, let $d$ be the greatest common divisor $d = gcd(a, b)$. Then there exists integers $x$ and $y$ such that

$$\begin{equation} \label{bezout} ax + by = d \end{equation}$$

If $a$ and $b$ are relatively prime then $gcd(a, b) = 1$ and by Bezout’s Identity there are integers $x$ and $y$ such that

$$ax + by = 1$$

Example: $3x + 8y = 1$, one solution is $x = 3$ and $y = -1$

## Extended Euclidean Algorithm

See divisibility for more details.

### Implementation

/**
* Computes the values x and y for the equation
*
*    ax + by = gcd(a, b)
*
* Given that a and b are positive integers
*
* @param {int} a
* @param {int} b
* @param {int} x
* @param {int} y
* @returns {int} gcd(a, b)
*/
int extended_euclidean(int a, int b, int &x, int &y) {
if (b == 0) {
x = 1;
y = 0;
return a;
}
int x1, y1;
int gcd = extended_euclidean(b, a % b, x1, y1);
x = y1;
y = x1 - a / b * y1;
return gcd;
}

/**
* Alternative version using a vector of ints
* Computes the values x and y for the equation
*
*    ax + by = gcd(a, b)
*
* @returns {vector<int>} A triplet with the values (gcd(a, b), x, y)
*/
vector<int> extended_euclidean(int a, int b) {
if (b == 0) {
// base case:
// b divides a so a(1) + b(0) = a
return vector<int> {a, 1, 0};
}
vector<int> t = extended_euclidean(b, a % b);
int gcd = t;
int x1 = t;
int y1 = t;
return vector<int> {gcd, y1, x1 - a / b * y1};
}


## Applications

### Diophantine equations

Equations with integer variables and coefficients are called Diophantine equations, the simplest non-trivial linear equation has the form

$$\begin{equation}\label{linear-diophantine-equation} ax + by = c \end{equation}$$

Where $a, b, c$ are given integers and $x, y$ are unknown integers

Using the extended Euclidean algorithm it’s possible to find $x$ and $y$ given that $c$ is divisible by $gcd(a, b)$ otherwise the equation has no solutions, this follows the fact that a linear combination of two numbers continue to be divided by their common divisor, starting with \eqref{bezout}

$$ax_g + by_g = gcd(a, b)$$

multiplying it by $\tfrac{c}{gcd(a, b)}$

$$\begin{equation}\label{diophantine-equation-gcd} a \cdot x_g \cdot \Big( \frac{c}{gcd(a, b)} \Big) + b \cdot y_g \cdot \Big( \frac{c}{gcd(a, b)} \Big) = c \end{equation}$$

then one of the solutions is given by

$$ax_0 + by_0 = c$$

where

$$\begin{cases} x_0 = x_g \cdot \big( \frac{c}{gcd(a, b)} \big) \\ y_0 = y_g \cdot \big( \frac{c}{gcd(a, b)} \big) \end{cases}$$

we can find all of the solutions replacing $x_0$ by $x_0 + \tfrac{b}{gcd(a, b)}$ and $y_0$ by $y_0 - \tfrac{a}{gcd(a, b)}$

$$a \cdot \Big( x_0 + \tfrac{b}{gcd(a, b)} \Big) + b \cdot \Big( y_0 - \tfrac{a}{gcd(a, b)} \Big) = ax_0 + \tfrac{ab}{gcd(a, b)} + by_0 - \tfrac{ab}{gcd(a, b)} = ax_0 + by_0 = c$$

This process could be repeated for any number in the form

$$\begin{cases} x = x_0 + k \cdot \big( \frac{b}{gcd(a, b)} \big) \\ y = y_0 - k \cdot \big( \frac{a}{gcd(a, b)} \big) \end{cases}$$

Where $k \in \mathbb{Z}$

/**
* Computes the integer values x and y for the equation
*
*    ax + by = c
*
* if c is not divisible by gcd(a, b) then there isn't a valid solution,
* otherwise there's an infinite number of solutions, (x, y) form one pair
* of the set of possible solutions
*
* @param {int} a
* @param {int} b
* @param {int} c
* @param {int} x
* @param {int} y
* @returns {bool} True if the equation has solutions, false otherwise
*/
bool linear_diophantine_solution(int a, int b, int c, int &x, int &y) {
int gcd = extended_euclidean(abs(a), abs(b), x, y);
if (c % gcd != 0) {
// no solutions since c is not divisible by gcd(a, b)
return false;
}
x *= c / gcd;
y *= c / gcd;
if (a < 0) { x *= -1; }
if (b < 0) { y *= -1; }
return true;
}


### Linear congruence equations

A linear congruence is a congruence $\pmod p$ of the form

$$ax \equiv b \pmod m$$

By the definition of the congruence relation $m \mid ax - b$

$$ax - b = my$$

Reordering the equation

$$ax - my = b$$

Which is a linear diophantine equation discussed above, it’s solvable only if $b$ is divisible by $gcd(a, m)$, additionally $gcd(a, m)$ tells us the number of distinct solutions in the ring of integers modulo $m$