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

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

Let \(a = dx\) and \(b=dy\) where x, y ∈ ℤ:

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