Proving two integers are relatively prime using Bezout's Theorem.

If $\operatorname{gcd}(a, b) = 1,$ $a$ and $b$ are relatively prime and also if $\operatorname{gcd}(a, b)$ is equal to $1$ there should be $2$ integers $k$ and $m$ such that $ak + bm = 1.$

If we can find such integers k and m, is it a proof that $a$ and $b$ are relatively prime ?

What you think about that proof? Is it correct way?

Solution 1:

$a, b$ be two integers and suppose there exists two integers $u, v$ such that $$au+bv=1$$

Claim: $a, b$ are relatively prime or co-prime.

Suppose, $ gcd(a,b)= d$

So, $d$ is a common divisor of $a$ and $b$ .

$d|a$ and $d|b$ .

Hence, $d|au+bv =1$

$d\in\{-1, 1\}$

As, $d$ is the greatest common divisor implies $d=1$.

Hence, $gcd(a, b) =1$