Solutions of Polynomial Congruences

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

Let $$a \equiv b \pmod n$$.

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

In particular, $$a$$ is a solution to the polynomial congruence $$P \left({x}\right) \equiv 0 \pmod n$$ iff $$b$$ is also.

Proof
Let $$P \left({x}\right) = c_m x^m + c_{m-1} x^{m-1} + \cdots + c_1 x + c_0$$.

Since $$a \equiv b \pmod n$$, from Modulo Multiplication and Congruence of Powers, we have $$c_r a^r \equiv c_r b^r \pmod n$$ for each $$r \in \Z: r \ge 1$$.

From Modulo Addition we then have:

$$ $$ $$

In particular, $$P \left({a}\right) \equiv 0 \pmod n$$ iff $$P \left({b}\right) \equiv 0 \pmod n$$.

That is, $$a$$ is a solution to the polynomial congruence $$P \left({x}\right) \equiv 0 \pmod n$$ iff $$b$$ is also.