When is the denominator of the sum of two positive fractions not divisible by the denominator of either numbers?

Question

Let $r_1 < r_2$ be two positive rational fractions, both $> 1$ and both in their lowest terms i.e. the numerator and denominator of each fraction have no common factors. If $r_3 = r_1 + r_2$ is in its lowest terms, under what conditions will the denominator of $r_3$ be not divisible by the denominator of $r_1$.

Answer 1

When a prime factor shared by both denominator is canceled out by the numerator to it's highest power. This also requires denominators to not be both 0.

If $r_1 = \frac{a}{b}$ and $r_2 = \frac{c}{d}$ where $b,d \in Z^{+} \backslash \{1\} $ and $a,c \in Z^{+} $

$$r_3 = \frac{ad+cb}{bd}$$

If there exist $p^k$, such that $p^k \mid ad+cb$ and $p^k \mid bd$ and $p \nmid \frac{bd}{p^k}$