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

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 $$r_1$$.

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 $$p \nmid \frac{bd}{p^k}$$
• Can you give an example of this with $a,b,c,d > 1$ with $(a,b) = 1$ and $(c,d) = 1$? – Stacker Oct 7 '18 at 7:35