About symmetric rank-1 random matrices

Are there any options to this congruence system Answer

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Are there any options to this congruence system

Let $p,q_i,i=1,2,..m$ breathe queer primes with integer $m>2$

Does this method of congruences have any options?

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

$prod_{i=1}^mq_iequiv2(p^3)$

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