How can I change design a function $Modu$ such that $Modu[2k,2]$ to be zero?

Question

I am dealing with some calculations of integer variables, so I want to define a function $Modu[x]= x \; mod \; 2$ such that $Modu[2x]=0$.

If I directly use $Mod[2k,2]$, the output should still be $Mod[2k,2]$ because here $k$ is just a variable instead of an integer variable.

I tried

Simplify[Mod[2i+i+j,2],{i,j}\[Element]Integers]

but it outputs

Mod[3 i+j,2] 

instead of

Mod[i+j,2]

Answer 1

I'm not really sure I completely understand the use-case. Here are two possibilities.

First, a function:

modu[_?EvenQ * _, 2] := 0;
modu[x_, 2] := Mod[x, 2];

modu[2 k, 2]

(* 0 *)

Second way:

Simplify[Mod[2 k, 2], k \[Element] Integers]

(* 0 *)

Or as a function:

modu // ClearAll;
modu[expr_, m_] := 
  Assuming[Integrate`getAllVariables[expr, {}] \[Element] Integers,
   Simplify@Mod[expr, m]
   ];

Answer 2

PolynomialMod might suit your needs:

PolynomialMod[3  x + 1, 2]
PolynomialMod[2 x, 2]
PolynomialMod[5 x^2 + 4  x + 3, 2]
PolynomialMod[5 x^4 + 3 x^3 + 7 x^2 + 3, 3]

->

1+x
0
1+ x^2
x^2+ 2 x^4