# Thread: Proving that no cancellation is possible in the fraction (a+c)/(b+d), if ab-cd=1

1. ## Proving that no cancellation is possible in the fraction (a+c)/(b+d), if ab-cd=1

Hi,

I have the following problem, but I don't know what to do. I have been told by someone to think about gcd(a+c, b+d), but I still don't know how to prove this. Thanks for your help!

"Prove that for any positive integers a, b, c, d satisfying

$\displaystyle ab-cd=1$

no cancellation is possible in the fraction

$\displaystyle (a+c)/(b+d)$, where the result is a rational number."

2. The set of rationals is closed under addition and nonzero division. Thus there is no way that the final expression you gave could be irrational.

3. I am sorry, I got confused with another exercise so I typed it wrong... It is supposed to be rational :-/

4. Perhaps start by solving the equation for a, and substituting to get $\displaystyle \frac{1+c(b+d)}{b(b+d)}$.

5. What you want to show is that $\displaystyle a+c, b+d$ are relatively prime.

It suffices to notice that $\displaystyle b(a+c)-c(b+d)=1$.