Is there any prime number $a$ satisfying $a=\frac{k_1-k_2}{k_1-bk_2}$?

Question

Is there $k_1, k_2, b$ positive integers, $\gcd(k_1,k_2)=1$, $k_1>k_2$, $b>1$ such that $$\frac{k_1-k_2}{k_1-bk_2}$$ is prime number?

Answer 1

Would $k_1=31, \ k_2= 3, b=9$ work?

$$ \frac{31-3}{31-9\cdot 3}=\frac{28}{4}=7. $$