Example

Define "addition modulo 5" and "multiplication modulo 5" in the following way: Define $a + b \pmod{5}$ to mean the remainder upon division by 5 of $a+b$, and define $ab \pmod{5}$ to be the remainder upon division by 5 of $ab$. Construct an addition table for numbers $0,1,2,3,4$ and a multiplication table for numbers $1,2,3,4$.


In both cases, we simply do the normal operation (addition or multiplication) and then find the remainder upon division by 5 of the result. This remainder is our answer. So for example:

$$4 \cdot 3 = 12$$
and 12 has remainder 2 upon divison by 5, so...
$$4 \cdot 3 \equiv 2 \pmod{5}$$
Likewise, if we look at addition...
$$4 + 2 = 6$$
and 6 has remainder 1 upon division by 5, so...
$$4 + 2 \equiv 1 \pmod{5}$$
Using the same technique, the addition and multiplication tables asked for are found very quickly.

The addition table$\pmod{5}$ is given by


$+$ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 0
2 2 3 4 0 1
3 3 4 0 1 2
4 4 0 1 2 3


The multiplication table$\pmod{5}$ is given by:

$\cdot$ 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1