In Section 3.4 we have encountered the addition of hours, weekdays, and months as an example for modular arithmetic. We now introduce binary operations on the sets $$\Z_n=\{0,1,2,\dots,n-1\}$$ where $$n\in\N$$ based on the addition and multiplication of integers. For $$a$$ and $$b$$ in $$\Z_n$$ we consider $$(a+b)\fmod n$$ and $$(a\cdot b)\fmod n\text{.}$$ Because the remainder of division by $$n$$ is always an element of $$\Z_n=\{0,1,2,\dots,n-1\}$$ these yield binary operations on $$\Z_n\text{.}$$

### Definition14.3.1.

Let $$n\in\N\text{.}$$ We define two binary operations on the set

\begin{equation*} \Z_n=\{0,1,2,\dots,n-1\}\text{.} \end{equation*}
1. We call $$\oplus:\Z_n\times\Z_n\to\Z_n\text{,}$$ $$a\oplus b:=(a+b)\fmod n\text{.}$$ addition modulo $$n$$.

2. We call $$\otimes:\Z _n\times\Z_n\to\Z_n\text{,}$$ $$a\otimes b:=(a\cdot b)\fmod n$$ multiplication modulo $$n$$.

We have already encountered operation tables for modular addition and multiplication Chapter 13. In Table 14.3.3 we present the operations tables for addition and multiplication modulo $$7$$ side by side. Once these tables are created modular addition or multiplication can be done by table lookup.

We present examples for addition and multiplication modulo 7. Let $$a\oplus b:=(a+b)\fmod 7$$ and $$a\otimes b:=(a\cdot b)\fmod 7\text{.}$$ Tables for the binary operations $$\oplus$$ and $$\otimes$$ are given in Table 14.3.3.

1. $$5\otimes 4=(5\cdot 4)\fmod 7$$ $$=20\fmod 7=6$$

2. $$3\oplus 4=(3+4)\fmod 7$$ $$==7\fmod 7=0$$

3. $$2\otimes(3\oplus 6)=2\otimes((3+6)\fmod 7)$$ $$=(2\otimes (9\fmod 7))$$ $$=2\otimes 2=(2\cdot 2)\fmod 7$$ $$=4\fmod 7=4$$

In Checkpoint 14.3.4 and Checkpoint 14.3.5 compute some modular sums and products.

Let $$\oplus:\mathbb{Z}_{97}\times\mathbb{Z}_{97}\to\mathbb{Z}_{97}$$ be defined by $$a\oplus b= (a +b) \bmod 97\text{.}$$

Compte

$$56\oplus 68 =$$

$$68\oplus 56 =$$

$$(81\oplus 90)\oplus 50 =$$

$$81\oplus (90\oplus 50) =$$

Let $$\otimes:\mathbb{Z}_{37}^\otimes\times\mathbb{Z}_{37}^\otimes\to\mathbb{Z}_{37}^\otimes$$ be defined by $$a\otimes b= (a \cdot b) \bmod 37\text{.}$$

Compute

$$11\otimes 31 =$$

$$31\otimes 11 =$$

$$19\otimes 20 =$$

$$20\otimes 19 =$$

$$(16\otimes 7)\otimes 9 =$$

$$16\otimes (7\otimes 9) =$$

$$(11\otimes 21)\otimes 18 =$$

$$11\otimes (21\otimes 18) =$$

In Theorem 3.4.10 and Theorem 3.4.14 we had seen that addition and multiplication and $$\fmod$$ work nicely together. These properties help make modular arithmetic easier as they help to keep the size of numbers small.

Let $$a:=9638\text{,}$$ $$b:=5920\text{,}$$ $$c:=30\text{,}$$ and $$d:=8423\text{.}$$

Compute:

$$a \bmod 10 =$$

$$b \bmod 10 =$$

$$c \bmod 10 =$$

$$d \bmod 10 =$$

Now use these to compute the following:

$$(a + b)\bmod 10 =$$

$$(c + d)\bmod 10 =$$

$$( b\cdot c)\bmod 10 =$$

$$( d\cdot a)\bmod 10 =$$

$$8$$

$$0$$

$$0$$

$$3$$

$$8$$

$$3$$
$$0$$
$$4$$
In the following two section we apply modular addition and multiplication in the definition of certain groups. We show that for any $$n\in\N\text{,}$$ the set $$\Z_n$$ with addition modulo $$n$$ is a group and that for any prime number $$p$$ the set $$\Z_p^\otimes$$ with multiplication modulo $$p$$ is a group.