Intersection between set of prime divisors

Question

Given a prime number $p>2$ and a finite set of primes $l_1,...,l_n$ (different from $p$). By Fermat's little theorem, we know that $l_1 | (p^{l_1-1}-1)$, so my questions are:

1. Is there an integer $r$ such that $l_i | (p^r-1)$ for all $i=1,...,n$?
2. Is there an integer $r$ such that $l_i \nmid (p^r-1)$ for all $i=1,...,n$?

Answer 1

1. Yes, $r=\prod_{i=1}^n (l_i-1)$.
2. Not in general: if $l\mid (p-1)$, then $l\mid (p^r-1)$ for all $r\in \Bbb N$.