# Solutions To Mathematics Textbooks/Algebra (9780817636777)/Exercises 26-50

Alternatively, you must essentially show that a and b are coprime; that is the numerator and denominator share no common factor. Another way of saying this is to say that ${\displaystyle gcd(a,b)=1}$.
Let ${\displaystyle g=gcd(a,b)}$. We can write ${\displaystyle ad-bc=\pm 1}$ as ${\displaystyle \left({\frac {a}{g}}\cdot gd-{\frac {b}{g}}\cdot gc\right)=g\left({\frac {a}{g}}d-{\frac {b}{g}}c\right)=\pm 1}$. Thus ${\displaystyle g}$ must be either -1 or 1, and thus a and b are coprime.