Definition:Polynomial Congruence

Definition
Let $$P \left({x}\right)$$ be an integral polynomial.

Then the expression:
 * $$P \left({x}\right) \equiv 0 \pmod n$$

is known as a polynomial congruence.

Solution
A solution of a polynomial congruence modulo $$n$$ is a residue class modulo $n$ such that any element of that class satisfies the congruence.

From Solutions of Polynomial Congruences, if one such element of a congruence class satisfies the congruence, they all do.

Number of Solutions
Let $$S = \left\{{b_1, b_2, \ldots, b_n}\right\}$$ be a complete set of residues modulo $m$.

The number of solutions of the congruence $$P \left({x}\right) \equiv 0 \pmod n$$ is the number of integers $$b \in S$$ for which $$P \left({b}\right) \equiv 0 \pmod n$$.