TITLE: Find odd primes $p$ and $q$ such that $(p-1)\mid {3q-1}$ and $(q-1)\mid{3p-1}$.
QUESTION [6 upvotes]: Find odd primes $p$ and $q$ such that $(p-1)\mid {3q-1}$ and $(q-1)\mid{3p-1}$.

My progress till now: I got $p=11$ and $q=17$ as a solution satisfying this question.
Can anyone give me some hints rather than a solution ? Thanks in advance .

REPLY [5 votes]: The condition tells us that $\frac{3p-1}{q-1}$,$\frac{3q-1}{p-1}$, and $\frac{3p-1}{p-1} \frac{3q-1}{q-1}$ is an integer. However, for $p,q\ge5$, we have
$$\frac{3p-1}{p-1} \frac{3q-1}{q-1} < \frac{3p}{\frac{4}{5}p} \times \frac{3q}{\frac{4}{5}q} <15$$