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. 

Hint: $(a+m)^n = a^n+\sum_{i=1}^nca^{n-i}m^i\equiv a^n $ mod $m$.
