Chapter 2.8, Problem 1TFE

Label each of the following statements as either true or false.

The notation $x=y$ mod $n$ is used to indicate the unique integer $x$ in the range $0\le x<n$ such that $x\text{ - }y$ is a multiple of $n$.

To determine

Whether the statement, “The notation $x=y\mathrm{mod}n$ is used to indicate the unique integer $x$ in the range $0\le x<n$ such that $x-y$ is a multiple of $n$.” is true or false.

Answer to Problem 1TFE

Solution:

The statement, “The notation $x=y\mathrm{mod}n$ is used to indicate the unique integer $x$ in the range $0\le x<n$ such that $x-y$ is a multiple of $n$.” is true.

Explanation of Solution

Consider the statement, “The notation $x=y\mathrm{mod}n$ is used to indicate the unique integer $x$ in the range $0\le x<n$ such that $x-y$ is a multiple of $n$.”

The congruence notation gives for a fixed $y$, $x\equiv y(\mathrm{mod}n)$ allows $x$ to be any integer such that $x-y$ is a multiple of $n$, but the notation $x=y\mathrm{mod}n$ requires $x$ to be the unique integer in the range $0\le x<n$ such that $x-y$ is a multiple of $n$.

Hence, the statement, “The notation $x=y\mathrm{mod}n$ is used to indicate the unique integer $x$ in the range $0\le x<n$ such that $x-y$ is a multiple of $n$” is true.