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

## Exercise 42Edit

### aEdit

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 ${\displaystyle gcd(a,b)=1}$ .

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.