Combining two consecutive integers starting with an even one yields one.

  xxxxxxx0 ^ xxxxxxx1 = 00000001 

If we start at a number divisible by four and do this twice, we get one twice.

  xxxxxx00 ^ xxxxxx01 = 00000001
  xxxxxx10 ^ xxxxxx11 = 00000001

And combining the two of course yields zero and we are right back at the start.