If we have two Polynomials $f(x)$ and $g(x)$ with integer coefficients such that $f(x) \equiv g(x)\left( {\bmod n} \right),n \in {\mathbb{Z}^ + }$
Does this mean that the coefficients of $f(x)$ are congruent coefficients of $g(x)$ mod n?
If it's True
Why ${x^{p}} \equiv x\left( {\bmod p} \right) \Rightarrow 1 \equiv 0\left( {\bmod p} \right)$?