First you can make every integers including the negative one to be 0,1,2,3, or 4 modulo 5.

For example

since -63=-65+2=5(-13)+2, then -63=2 modulo 5.

Then one you translate every integer into either 0,1,2,3 or 4 then addition and multiplication are basically addition and multiplication involving 0,1,2,3,4 only.