1. ## [SOLVED] addition and multiplication tables

I have these two Cayley tables. The addition table is complete and the multiplication table is not. The instructions say to use the distributive laws, but I can't seem to find the solutions using the distributive laws. I can only find the solution using the definition of a group.

Can someone help me?

ring R = {a, b, c}

+ a b c
a a b c
b b c a
c c a b

x a b c
a a a a
b a c
c a

Thanks a bunch!

2. The additiion table shows that $\langle R+\rangle$ is a group with identity element $a.$ This is the zero element of $R.$

To complete the table for multiplication, take for example $bc.$ From the addition table, $c=b+b.$ Hence, $bc=b(b+b)=bb+bb=c+c=b.$ (You know $bb=c$ from the partially completed table.)

Do the same for $cb$ and $cc.$ (Hint: It will turn out that $c$ is the multiplicative identity, and the ring is a field with 3 elements.)