If two fractions can't be canceled further and their denominators are coprime then also their sum can't be canceled further

Question

The fractions $\frac{a}{b}$, $\frac{c}{d}$ can't be canceled further and their denominators $b, d$ are coprime. Then also the sum of the fractions $\frac{ad+bc}{bd}$ can't be canceled further.

So we know that $\gcd(b,d)=1$. And since the fractions can't be canceled further I assume that $\gcd(a,b)=\gcd(c,d)=1$. However I'm unsure on how to proceed from here.

score 1 · Answer 1 · answered Mar 29 '23 at 15:55

1

Any prime factor of (e.g.) $b$ that divides $ad+bc$ must also divide $ad$. Can you take it from here?

answered Mar 29 '23 at 15:55

Torsten Schoeneberg

26,051

Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Mar 29 '23 at 16:39

If two fractions can't be canceled further and their denominators are coprime then also their sum can't be canceled further

1 Answers1