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Chapter 4: Problem 1

This question concerns congruence mod 7 . (a) List three positive and three negative integers in \(\overline{5}\) and in \(\overline{-3}\). (b) What is the general form of an integer in \(\overline{5}\) and of an integer in \(\overline{-3}\) ?

Short Answer

Expert verified
See individual steps for lists; forms are \( 5 + 7k \) and \( 4 + 7k \).

Step by step solution

01

Understand Congruence Classes

In modular arithmetic, the notation \( \overline{5} \) represents the set of all integers that, when divided by 7, leave a remainder of 5. Similarly, \( \overline{-3} \) represents integers that leave a remainder of -3, but we usually want positive remainders so we convert this to \( 7 - 3 = 4 \). Thus \( \overline{-3} = \overline{4} \).
02

List Positive Integers in \(\overline{5}\)

Integers in \(\overline{5}\) are of the form \( 5 + 7k \), where \( k \) is any integer. By choosing \( k = 0, 1, 2 \), we get the positive integers 5, 12, and 19.
03

List Negative Integers in \(\overline{5}\)

For negative integers, choose \( k = -1, -2, -3 \) to yield the integers -2, -9, and -16.
04

List Positive Integers in \(\overline{-3}\) Equivalent to \(\overline{4}\)

For \( \overline{4} \), use the formula \( 4 + 7k \). Choosing \( k = 0, 1, 2 \), we get the positive integers 4, 11, and 18.
05

List Negative Integers in \(\overline{-3}\) Equivalent to \(\overline{4}\)

Using \( k = -1, -2, -3 \) in \( 4 + 7k \), we find the negative integers -3, -10, and -17.
06

Define General Forms

The general form of integers in \(\overline{5}\) is \( 5 + 7k \), and for \(\overline{-3} \) (which is \(\overline{4}\)), it is \( 4 + 7k \), where \( k \) is any integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Congruence Classes
Congruence classes are a fundamental concept in modular arithmetic. Imagine organizing numbers into groups based on their remainders when divided by a fixed number, known as the modulus. Each group is a congruence class.

For example, the notation \( \overline{a} \) refers to a congruence class of integers that have a specific remainder \( a \) when divided by the modulus. In our exercise, \( \overline{5} \) and \( \overline{-3} \) represent such classes with modulus 7.
  • \( \overline{5} \) includes numbers like 5, 12, and 19, each having a remainder of 5 when divided by 7.
  • \( \overline{-3} \) is equivalent to \( \overline{4} \) because a negative remainder like -3 becomes positive by adding the modulus: 7 - 3 = 4.
By understanding congruence classes, one can easily categorize and work with integers in modular arithmetic.
Integers
Integers are whole numbers that can be positive, negative, or zero. In modular arithmetic, we often talk about the general form of integers in a congruence class.

For instance, consider the integers in \( \overline{5} \) under modulo 7. These integers can be expressed in a general form: \( 5 + 7k \), where \( k \) is any integer. This formula tells us that to find any integer in \( \overline{5} \), start at 5 and add multiples of 7.
  • For positive \( k \), choosing values like 0, 1, 2 results in 5, 12, and 19.
  • For negative \( k \), such as -1, -2, -3, it results in -2, -9, and -16.
These patterns make it simple to identify and generate other members of a congruence class of integers easily.
Remainders
Remainders allow us to classify numbers based on what is leftover when one number is divided by another. It's a crucial concept in modular arithmetic.

When working with modulus 7, we divide integers by 7 and focus on the remainder. For example:
  • The remainder of 12 divided by 7 is 5, thus 12 belongs to \( \overline{5} \).
  • When dealing with a negative number like -3, the remainder is adjusted to a positive equivalent by adding the modulus. Therefore, \(-3 + 7 = 4\).
This conversion to a positive remainder ensures that every member of a congruence class has a consistent and easily understandable label.
Mod 7
The term 'mod 7' refers to an arithmetic system where numbers are divided by 7, and we focus on the remainders. It's a way to cycle through a small range of numbers.

When performing calculations mod 7, only the remainders 0 through 6 are possible. They represent all possible outcomes of division by 7.
  • Positive integers like 4 and 5 result in remainders \( 4 \) and \( 5 \), part of classes \( \overline{4} \) and \( \overline{5} \).
  • Negative numbers, such as -3, are converted to \( \overline{4} \) because mod 7: \(-3 \equiv 4 \mod 7\).
Understanding mod 7 calculations helps in simplifying and solving problems involving remainders and congruence classes.

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