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Let $q$ be an odd prime and $f(x)$, $g(x)$ be polynomials with integercoefficients. If the system of congruences $f(x) \equiv g(x) \equiv 0 \pmod{q}$has $\ell$ solutions, then $R\left(f(x),g(x)\right)\equiv 0 \pmod{q^\ell}$,where $R\left(f(x),g(x)\right)$ is the resultant of the polynomials. Using thisresult we give new proofs of some known congruences involving the Lucassequences.

On the relationship between the number of solutions of congruence systems and the resultant of two polynomials