Proof

Let $a$ and $b$ be integers with $d=\gcd(a,b)$.
If $f > 0$ is a common divisor of $\frac ad$ and $\frac bd$, then it follows from the divisibility law no. 8 that $fd\mid a,$ and $fd\mid b.$
Therefore, from the definition of the greatest common divisor, we must have $fd\mid d,$ and $fd\mid d.$
Applying divisibility law no. 7, it follows that $f\mid 1,$ and therefore $f=1.$
By definition, the integers $\frac ad$ and $\frac bd$ are co-prime.
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Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927