Simplifying a Modular Arithmetic Equation

Question

Is there a theorem or rule that makes this equivalent modular arithmetic equation in a form of: $$f(x) = (a + b)(x \ \mathrm{mod}\ 2 )$$ into something like this: $$f(x) = (ax \ \mathrm{mod}\ 2 ) + (bx \ \mathrm{mod}\ 2 )$$ which later can be simplify using distributive law.

Answer 1

Your first statement says that $$f(x)=(a+b)(x+2p)=ax+bx+2pa+2pb$$ while the second states that $$f(x)=2m+ax+2n+bx$$ for some $p,m,n$ in $\mathbb{Z}.$

The equality will hold under very special conditions. Can you take it from here?