I was asked this question in an exam
For an integer $n>3$ denote by $f(n)$ the product of all prime numbers less than $n$. So $f(6) = 30$, $f(5) = 6$. Which of the following are true?
A. There are only finitely many $n$ such that $n|f(n)$.
B. There are only finitely many $n$ such that $n>f(n)$.
C. Given any $k,l \in \mathbb N$, $f(n)=kn+l$ has infinitely many solutions.
D. Let $S_k=\{l\in \mathbb N : f(l)=k\}$. Let $a_k=\frac 1 {|S_k|}$ if $S_k\neq\phi$, else $a_k=0$. Then $\sum_{i=1}^\infty a_i$ does not converge to a rational number.
Now, for A, I figured out an idea. Consider the set of the first $k$ primes $P_k=\{p_i : i\in \mathbb N_k \}$ where $\mathbb N_k=\{1,2,\dots ,k\}$ and let the $(k+1)$-th prime be $p_{k+1}$. Now consider the set of all possible products in $P_k$ given by $\prod P_k=\left\{\prod_{i=1}^m p_{\pi (i)} : p_j\in P_k, m\in \mathbb N_k, \pi \text{ is any permutation of }\mathbb N_k \right\}$. Now, if there is at least one element $x\in\prod P_k$ which is in the interval $(p_k,p_{k+1})$, then it is trivial that $x|f(x)$. And, I guess, there will always be such an $x$. However, a proof of this, or a better way will be appreciated.
For B, it is clear that $f(n)$ increases much much faster than $n$. So, it's no doubt that it's true. Again, I would like to have a rigorous proof of this.
For C, if we put $k=l=1$, we can intuitively see that $f(n)=n+1$ will not have infinitely many solutions (since $f(n)$ increases much much faster than $n+1$). Again, a proof of this will be appreciated.
My main problem was with D. I can figure out that $|S_k|\neq \phi$ iff $k$ is of the form $\prod_{i=1}^m p_i$ where $p_1,p_2,\dots ,p_m$ are the first $m$ primes. So, $a_k$ will either be $0$ (when $|S_k|=\phi$), or $a_k$ will be equal to the number of integers between $(p_{m}+1)$ and $p_{m+1}$. But, I can't see how to relate it with whether the sum will converge to a rational number or not. Please help me in these.
Edit: The answers of this question (according to the answer key) are B and D, all of which are done either on the comments or in the answers!!! Thanks to you all :)