Consider congruences of the form

(1)

• If $a_i \equiv 0 \pmod{m}$ for all $i$ 's, the congruence is satisfied by each integer, in which case the congruence is identical (cf. the formal congruence). Therefore one can assume that at least $$a_n \not\equiv 0 \pmod{m},$$ since one would otherwise have $a_nx^n \equiv 0 \pmod{m}$ and the first term could be left out of (1). Now, we say that the degree of the congruence (1) is $n$ .
• If $x = x_0$ is a solution of (1) and $x_1 \equiv x_0 \pmod{m}$ , then by the properties of congruences, $$f(x_1) \;\equiv\; f(x_0) \;\equiv\; 0 \pmod{m},$$ and thus also $x = x_1$ is a solution. Therefore, one regards as different roots of a congruence modulo $m$ only such values of $x$ which are incongruent modulo $m$ .
• One can think that the congruence (1) has as many roots as is found in a complete residue system modulo $m$ .