# Bezout's Theorem

Bezout was a famous mathematician who discovered many beautiful formulas. One of his most famous theorem/identities dealt with numbers and their greatest common factors. His theorem states, if integers $$a$$ and $$b$$ are relatively prime, then there exists $$x$$ and $$y$$, integers to satisfy the equation $$ax+by=1$$. For any integers $$a$$ and $$b$$ excluding zero, let $$d$$ be the greatest common divisor such that $$d$$ equals the gcd of $$a$$ and $$b$$ in other words, $$d$$$$=$$gcd$$\left(a,b\right)$$. If this is true then there must exist integers known as $$x$$ and $$y$$ to satisfy the equation $$ax+by=d$$.

There are many important facts that go along with Bezout’s identity:

1.)    All common divisors of $$d$$ are common divisors of $$a$$ and $$b$$ as well
2.)    As for the fact above, all divisors of $$a$$ and $$b$$ are also divisors of $$d$$
3.)    $$a$$/$$d$$ and $$b$$/$$d$$ are prime integers
4.)    $$a$$/$$d$$$$\left(x\right)$$+$$b$$/$$d$$$$\left(y\right)$$$$=1$$
5.)    The greatest common divisor $$d$$ is actually the smallest integer that can be written to satisy $$ax+by$$
6.)    All integers of the form $$ax+by$$ are multiples of $$d$$

Next we will work out an example.

If we have ﻿$$2x+3y=1$$, well by trial and error we can quickly see that ﻿$$x$$ is ﻿$$2$$ while ﻿$$y$$ is ﻿$$-1$$. But when we are given huge numbers it can be a bit more difficult. In order to work the equation out with nigger numbers the Euclidean algorithm must be applied. Let’s recall that Euclid’s theorem states: For any numbers and ﻿$$a,b,$$ ﻿$$x,$$ gcd ﻿$$\left(a,b\right)$$= gcd﻿$$\left(b,a-bx\right)$$