I had been researching over the Extended Euclidean Algorithm when I happened to observe that the Bézout Coefficients were always relatively prime.
Let $a$ and $b$ be two integers and $d$ their GCD. Now, $d = ax + by$ where x and y are two integers.
$$d = ax + by \implies 1 = \frac{a}{d}x + \frac{b}{d}y$$ So, $x$ and $y$ can be expressed to form 1 so their GCD is 1 and are relatively prime. ($\frac{a}{d}$ and $\frac{b}{d}$ are integers.)
Another great thing is that $\frac{a}{d}$ and $\frac{b}{d}$ are also relatively prime. So you see this goes on like a sequence till $a$ and $b$ become one.
Am I right? What else can be known from this fact? Is it useful? Can it be used to prove some other things?