Efficient way to show the Associativity of $\oplus: E \times E \rightarrow E,\ x\oplus y \equiv x+y \bmod 5$ Let $E:=\{0, 1, 2, 3, 4\}$ be a set and let $\oplus$ be an internal binary operation on $E$ such that 
$$\oplus: E \times E \rightarrow E,\ x\oplus y \equiv x+y \bmod 5$$
So, we have:
\begin{array}{|c|c|c|c|c|c|c|}\hline
\oplus & 0 & 1 & 2 & 3 & 4 \\ \hline
0 & 0 & 1 & 2 & 3 & 4 \\  \hline
1 & 1 & 2 & 3 & 4 & 0 \\ \hline
2 & 2 & 3 & 4 & 0 & 1 \\ \hline
3 & 3 & 4 & 0 & 1 & 2 \\ \hline
4 & 4 & 0 & 1 & 2 & 3 \\ \hline
\end{array}
I need to conclude the associativity of $\oplus$ on $E$ by using the above table.
Is there an efficient way to conclude that without going through all $5\cdot5\cdot5= 125$ possibilities?

If you want to prove the associativity "using the table" yo have to do all $125$ cases, or else have to give a law describing the buildup of the table.
What I want to say: Your problem can be solved only by doing the $125$ cases using the associativity in ${\mathbb Z}$ individually, or by proving once and for all "abstractly" that the addition in ${\mathbb Z}_5$ (or a similar quotient structure) is again associative. Then your table just has to do with the naming of the equivalence classes.
