In the context of the study of pseudorandom sequences that use quadratic residues modulo the prime $p$, the constructive description of the set of prime moduli for which given integers are quadratic residues is considered. Using the Gauss Lemma, we prove a criterion of combinatorial nature for a given integer $a$ to be a quadratic residue prime modulo $p$. It is shown how to apply this criterion to the problem of effective description of the prime moduli $p$ satisfying the equation $\bigl(\frac ap\bigr)=1$ for each $p$ from a given finite set $M$.