Polynomials and Modular Arithmetic: Elementary Number Theory Problem Let $f(x)$ be a non-constant polynomial with integer coefficients. Let $a, k$, and $m$ be integers with $m > 1.$ Suppose that $f(a) ≡ k$ $\pmod  m.$ Prove that $f(a + m) ≡ k$ $\pmod  m.$
I've fiddled around, expanding $f(a + m)$, factoring etc., and am looking at $f(a + m) - f(a)$, but can't quite get it right. 

You can prove it  only on the monomial $x^r$
$(x+m)^r≡\sum_{i=0}^r\binom rix^{r-i}m^i≡x^r  mod(m)$.
