Chapter 10, Problem 1E

To determine

To compute: The addition and multiplication table for the integers mod 4.

Answer to Problem 1E

The addition and multiplication table for the integers mod 4 are:

$\begin{array}{ccccc}\oplus & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 1& 2& 3& 0\\ 2& 2& 3& 0& 1\\ 3& 3& 0& 1& 2\end{array}$ and $\begin{array}{ccccc}\otimes & 0& 1& 2& 3\\ 0& 0& 0& 0& 0\\ 1& 0& 1& 2& 3\\ 2& 0& 2& 0& 2\\ 3& 0& 3& 2& 1\end{array}$.

Explanation of Solution

Definition used:

“Let n be a positive integer with $n\ge 2$, then $Z_n=\left\{0,1,\dots ,n-1\right\}$.”

“For any two integers a and b in $Z_n$, $a\oplus b$ is the unique remainder when the ordinary sum $a+b$ is divided by n, and $a\otimes b$ is the unique remainder when the ordinary product $a\times b$ is divided by n.”

Calculation:

The addition and multiplication table for the integers mod 4 is to be calculated.

Thus, $n=4$.

By the definition of $Z_n$, the set $Z_4$ will have four elements 0, 1, 2 and 3. That is, $Z_4=\left\{0,1,2,3\right\}$.

Add 0 with each element of $Z_4$ and obtain the remainder when divided by 4.

$\begin{array}{l}0\oplus 0=0\\ 0\oplus 1=1\\ 0\oplus 2=2\\ 0\oplus 3=3\end{array}$

Add 1 with each element of $Z_4$ and obtain the remainder when divided by 4.

$\begin{array}{l}1\oplus 0=1\\ 1\oplus 1=2\\ 1\oplus 2=3\\ 1\oplus 3=0\end{array}$

Add 2 with each element of $Z_4$ and obtain the remainder when divided by 4.

$\begin{array}{l}2\oplus 0=2\\ 2\oplus 1=3\\ 2\oplus 2=0\\ 2\oplus 3=1\end{array}$

Add 3 with each element of $Z_4$ and obtain the remainder when divided by 4.

$\begin{array}{l}3\oplus 0=3\\ 3\oplus 1=0\\ 3\oplus 2=1\\ 3\oplus 3=2\end{array}$

The remainders form the addition table for the integers mod 4 as follows:

$\begin{array}{ccccc}\oplus & 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 1& 2& 3& 0\\ 2& 2& 3& 0& 1\\ 3& 3& 0& 1& 2\end{array}$

Similarly, multiply each elements of $Z_4$ and divided by 4 as follows:

$\begin{array}{l}0\otimes 0=0\\ 0\otimes 1=0\\ 0\otimes 2=0\\ 0\otimes 3=0\end{array}$

Multiply 1 with each element of $Z_4$ and obtain the remainder when divided by 4.

$\begin{array}{l}1\otimes 0=0\\ 1\otimes 1=1\\ 1\otimes 2=2\\ 1\otimes 3=3\end{array}$

Multiply 2 with each element of $Z_4$ and obtain the remainder when divided by 4.

$\begin{array}{l}2\otimes 0=0\\ 2\otimes 1=2\\ 2\otimes 2=0\\ 2\otimes 3=2\end{array}$

Multiply 3 with each element of $Z_4$ and obtain the remainder when divided by 4.

$\begin{array}{l}3\otimes 0=0\\ 3\otimes 1=3\\ 3\otimes 2=2\\ 3\otimes 3=1\end{array}$

The remainders form the multiplication table for the integers mod 4 as follows:

$\begin{array}{ccccc}\otimes & 0& 1& 2& 3\\ 0& 0& 0& 0& 0\\ 1& 0& 1& 2& 3\\ 2& 0& 2& 0& 2\\ 3& 0& 3& 2& 1\end{array}$