Let 
be any integer. Then
In other words:
(1) If is a multiple of 4, then 
.
(2) If dividing by 4 leaves a remainder of 1, then 
, since 
for some integer 
, and from (1) we know that 
(because, obviously, 
is a multiple of 4).
(3) If instead you get a remainder of 2, then
. This follows from (1) as well. 
.
(4) Finally, if you get a remainder of 3, then
.
In other words:
(1) If
(2) If dividing
(3) If instead you get a remainder of 2, then
(4) Finally, if you get a remainder of 3, then