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 $(\frac{a}{g}\cdot gd-\frac{b}{g}\cdot gc)=g(\frac{a}{g}d-\frac{b}{g}c)=\pm 1$. Thus $g$ must be either -1 or 1, and thus a and b are coprime. Template:BookCat