Solutions of Polynomial Congruences

Theorem
Let $\map P x$ be an integral polynomial.

Let $a \equiv b \pmod n$.

Then $\map P a \equiv \map P b \pmod n$.

In particular, $a$ is a solution to the polynomial congruence $\map P x \equiv 0 \pmod n$ $b$ is also.

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

Since $a \equiv b \pmod n$, from Congruence of Product 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, $\map P a \equiv 0 \iff \map P b \equiv 0 \pmod n$.

That is, $a$ is a solution to the polynomial congruence $\map P x \equiv 0 \pmod n$ $b$ is also.