(related to Proposition: Generating Co-Prime Numbers Knowing the Greatest Common Divisor)

- 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.∎

**Landau, Edmund**: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927