Turing machines for equation with modulo

I'm required to create a touring machine for

Z =(Xi + Ki)mod 2

but I'm completely lost in terms of creating a Turing machine for a modulo of 2. X and K are binary inputs where i is the length of the string. The input is given as such where:

XYK

the Y just acts as a separator for binary strings X and K which could vary in length. The problem I'm having now is regarding the modulo part of the equation. How do i begin with mod 2 and what I'm supposed to look out for?

Solution

Based on this I think what you are asking for is Z such that Z_i = X_i + Y_i (mod 2):

 (X0 X1 X2 ... Xi
+ K0 K1 K2 ... Ki)
%  2  2  2 ...  2
= Z0 Z1 Z2 ... Zi

Given this and an input tape like BXX...XY...KK...KBB... where B is blank, XX...X is an i-digit binary number, Y is a separator and KK...K is another i-digit binary number, the problem is easy:

Write a new separator V to the first non-blank cell after the input. Make sure you can distinguish it from X, Y, K. Return to the beginning of the tape.
Move right until you find a 0 or 1 belonging to X (if you find Y, skip below). Enter state X1 if 1 and X0 of 0. Write W on the tape and move right to the first 0 or 1 after Y. If you found Y before a 0 or 1, then copy V to the front of the tape, then write blanks over everything else and halt-accept.
If in state X0 and you see a 0, or if you're in X1 and you see a 1, enter state Z0. Otherwise, enter state Z1.
Write a W to the tape and move right to the first blank cell after V.
Write 0 if in Z0, or 1 if in Z1.
Move back to the beginning of the tape and repeat the process until you find Y first in step 2.

Example: 0011 + 1010

B0011Y1010BBBBB...
^

B0011Y1010VBBBB...
^                 move to the end of input, write V separator, reset head

B0011Y1010VBBBB...
 ^                move right to first 0

BW011Y1010VBBBB...
      ^           enter X0, write W, move right to first 1 after Y

BW011YW010VBBBB...
           ^      enter Z1, write W, move right to first blank after V

BW011YW010V1BBB...
^                 write 1, return to beginning, repeat

BWW11YWW10V10BB...
^                 find 0, X0, find 0, Z0, write 0, return to start, repeat


BWWW1YWWW0V100B...
^                 find 1, X1, find 1, Z0, write 0, return to start, repeat

BWWWWYWWWWV1001...
^                 find 1, X1, find 0, Z1, write 1, return to start, repeat

B1001BBBBBBBBBB...
^                 find Y, copy from after V to beginning, erase rest, halt.