  Let $q$ be an odd prime and $f(x)$, $g(x)$ be polynomials with integer
coefficients. 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 this
result we give new proofs of some known congruences involving the Lucas
sequences.
