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

Exercise 42

a

If a and b had a nontrivial common factor k >= 2, then a = k*a' and b = k*b', so (ad - bc) = k(a'd - b'c) = ±1.

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 $gcd(a,b)=1$.

Let $g=gcd(a,b)$. We can write $ad-bc=\pm 1$ as $\left({\frac {a}{g}}\cdot gd-{\frac {b}{g}}\cdot gc\right)=g\left({\frac {a}{g}}d-{\frac {b}{g}}c\right)=\pm 1$. Thus $g$ must be either -1 or 1, and thus a and b are coprime.