Problem 2 Show that if $x$ is an odd int... [FREE SOLUTION]

Chapter 3: Problem 2

Show that if $x$ is an odd integer, then $x^{2} \equiv 1(\bmod 8)$

Short Answer

Expert verified

Question: Prove that when x is an odd integer, $x^{2} \equiv 1(\bmod 8)$. Answer: If x is an odd integer, it can be represented in one of four possible forms: $x = 8a + 1, x = 8a + 3, x = 8a + 5, x = 8a + 7$. When squaring these forms, we obtain expressions that all result in a multiple of 8 when subtracting 1. This demonstrates that $x^{2} \equiv 1(\bmod 8)$ when x is an odd integer.

Step by step solution

Understanding Odd Integers

An odd integer is a number that can't be divided by 2 without leaving a remainder. In other words, an odd integer can be written as $2n+1$, where n is an integer.

Possible Forms of x

Since $x$ is an odd integer, it can be written in the general form as $x = 2n+1$. By observing the remainder when dividing by 8, we can claim that x can take one of the following forms: $$x = 8a + 1, x = 8a + 3, x = 8a + 5, x = 8a + 7$$ for some integer $a$.

Squaring the Possible Forms of x

Now we will square each of the possible forms of x and simplify: $$ (8a + 1)^2 = 64a^2 + 16a + 1 $$$$ (8a + 3)^2 = 64a^2 + 48a + 9 $$$$ (8a + 5)^2 = 64a^2 + 80a + 25 $$$$ (8a + 7)^2 = 64a^2 + 112a + 49 $$

Showing Congruence to 1 modulo 8

To prove that each squared form is congruent to 1 modulo 8, we just need to show that $x^2 - 1$ is divisible by 8 (i.e., it is a multiple of 8). Let's examine each case: $$ Case\: 1: (64a^2 + 16a + 1 - 1) = 64a^2 + 16a \equiv 0 \pmod 8 $$$$$ Case\: 2: (64a^2 + 48a + 9 - 1) = 64a^2 + 48a + 8 \equiv 0 \pmod 8 $$$$$ Case\: 3: (64a^2 + 80a + 25 - 1) = 64a^2 + 80a + 24 \equiv 0 \pmod 8 $$$$$ Case\: 4: (64a^2 + 112a + 49 - 1) = 64a^2 + 112a + 48 \equiv 0 \pmod 8 $$

Conclusion

All four possible squared forms of $x$ result in an expression that is a multiple of 8 when subtracting 1. Therefore, we have shown that if x is an odd integer, then $x^{2} \equiv 1(\bmod 8)$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Odd Integers

Odd integers are numbers that cannot be evenly divided by 2. This means when you divide an odd integer by 2, there is always a remainder of 1. For example, numbers like 1, 3, 5, 7, and so on are odd integers.

In mathematical terms, we can express any odd integer in the form:

$ x = 2n + 1 $

where $ n $ is an integer.This form helps us understand and explore their properties, especially when dealing with modular arithmetic and other aspects of number theory.

Congruences

In mathematics, congruences are a way to express that two numbers have the same remainder when divided by a number called the modulus. We write it as:

$ x \equiv y (mod n) $

This reads as $ x $ and $ y $ leave the same remainder when divided by $ n $.

Congruences are extensively used in number theory, particularly in problems involving divisibility and modular arithmetic. In our example, when we state that $ x^2 \equiv 1 \, (\bmod \, 8) $, we mean that when you square an odd integer, its remainder when divided by 8 will be 1.

Number Theory

Number theory is a vast field of mathematics focusing mainly on properties of integers. It includes various subfields such as arithmetic, algebraic number theory, and analytic number theory.

Topics like divisibility, prime numbers, and congruences play a vital role here.

In our exercise, number theory helps us analyze patterns and properties of numbers when dealing with operations like squaring odd integers. Notice how number theory simplifies proving properties like $ x^2 \equiv 1 \, (\bmod \, 8) $ by breaking down integers into manageable forms.

Integer Properties

Integers have several essential properties which make them unique among numbers. They are whole numbers that include negative infinity to positive infinity, but not fractions or decimals.

Some crucial properties of integers help solve or verify mathematical proofs:

Closure: The sum or product of two integers is always an integer.
Consistency: Integers follow a predictable pattern, simplifying calculations.
Behavior under operations: Understanding how odd, even, negative, and positive integers behave under addition, subtraction, multiplication, and division helps solve complex problems.

In our example, knowing how odd integers behave when squared and divided by 8 allows us to demonstrate modular congruences logically and systematically.

91影视

Show that if \(x\) is an odd integer, then \(x^{2} \equiv 1(\bmod 8)\)

Short Answer

Step by step solution

Understanding Odd Integers

Possible Forms of x

Squaring the Possible Forms of x

Showing Congruence to 1 modulo 8

Conclusion

Key Concepts

Odd Integers

Congruences

Number Theory

Integer Properties

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Applied Mathematics

Discrete Mathematics

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.