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 $P \left({b}\right) \equiv 0 \pmod n$.