For relatively prime numbers $n$, let $ord_m(n)$ denote the order of $n$ modulo $m$, that is, the smallest number $k$ such that $m\mid(n^k-1)$. Now let $q$ and $r$ be two distinct prime numbers. Dose there exists a prime number $p$ such that $ord_p(q)<ord_p(r)$?

(Note: this question can be asked even if $q$ and $r$ aren't prime, as long as we avoid situations like $r=q^2$.)