If d = gcd(a, b), then 1 = gcd(a/d, b/d)

Since $d =gcd(a, b)$, then $d$ is also the smallest positive linear combination of $a$ and $b$ (click here to see why):

$$d = ha + kb$$

Let $a = dx$ and $b=dy$ where x, y ∈ ℤ:

$$\begin{gathered} d = hdx + kdy\\ 1 = hx + ky \end{gathered}$$

This shows that $gcd(x, y) = 1$, or in other words, $gcd(a/ d, b/d) = 1$.