Given $2$ arithmetic progressions, $\{an+b\}_{n=0}^{\infty}$ and $\{cm+d\}_{m=0}^{\infty}$, where $a,c$ are distinct primes. These intersect any time there are integers $(n,m)$ such that $an+b=cm+d$, and after a single intersection $an_0+b=cm_0+d$ they will continue to intersect at $a(n_0+ct)+b=c(m_0+at)+d$ for every natural number $t$. The first intersection can be efficiently found using the extended Euclidean algorithm.
Now let's say we have $k$ such arithmetic progressions of the form $\{an+b\}_{n=0}^{\infty}$ with distinct values of $a$, where $a$ is a prime and $0<=b<a$. Also it is given that $a\approx L$.
What is the number of intersections in a single point, we should expect to see in a range of $0-R$ where $R$ is a natural number?
Answering this question will help my research on a new factoring method.