Thread: Field in Mod 10

Let R = {0,2,4,6,8} under addition and multiplication in Mod 10. Prove that R is a field.

My proof so far:

I understand that I need to prove there exist a unity in R, and for all nonzero elements in R, there exist their inverses.

Now, I found that 6 is the unity because for all elements a in R, 6a = a in Mod 10.

However, I have a bit of difficulty prove all elements are units.

Little help?

However, I have a bit of difficulty prove all elements are units.

All non-zero elements are units. First the identity element is 6. Next for every non-zero element, say 2, we need to show it has an inverse. That is, 2*x = 6 (because 6 is unity). So what does 2 multiply with to give 6 under mod 10. Then do this for the others.