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

aa b c

bb c a

cc a b

x a b c

aa a a

ba c

ca

Thanks a bunch!