Are there any solutions to this congruence system

Question

Let $p,q_i,i=1,2,..m$ be odd primes with integer $m>2$

Does this system of congruences have any solutions?

$\prod_{i=1}^m(q_i-1)\equiv2(p^2)$

$\prod_{i=1}^mq_i\equiv2(p^3)$

Answer 1

We can even get the first congruence hold modulo $p^3$ as well.

For example, $m=3$, $p=5$, $q_1=67$, $q_2=367$, and $q_3=743$.