/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Let \(a\) and \(b\) be integers.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 8

Let \(a\) and \(b\) be integers. Prove that if \(a \equiv 2(\bmod 3)\) and \(b \equiv 2(\bmod 3)\), then (a) \(a+b \equiv 1(\bmod 3) ;\) (b) \(a \cdot b \equiv 1(\bmod 3)\).

Short Answer

Expert verified
Using the given congruences \(a \equiv 2 (\bmod 3)\) and \(b \equiv 2 (\bmod 3)\), we rewrite \(a\) and \(b\) as \(a = 3x + 2\) and \(b = 3y + 2\) for some integers \(x\) and \(y\). We then find their sum and product: \(a+b = 3(x+y)+4 = 3(x+y+1)+1\) and \(a \cdot b = (3x+2)(3y+2) = 3(3xy + 2(x+y)+1) + 1\). Since 3 divides both \(3(x+y+1)\) and \(3(3xy + 2(x+y)+1)\), we can conclude that \(a+b \equiv 1 (\bmod 3)\) and \(a \cdot b \equiv 1 (\bmod 3)\), thus proving parts (a) and (b).

Step by step solution

01

Use the definition of modular congruence

Since \(a \equiv 2 (\bmod 3)\) and \(b \equiv 2 (\bmod 3)\), there exist integers \(x\) and \(y\) such that \(a=3x+2\) and \(b=3y+2\) by definition of modular congruence.
02

Find the sum using the given expressions for a and b

To prove that the sum has a remainder of 1 when divided by 3, we need to show that \(a+b \equiv 1 (\bmod 3)\). Using the expressions given above for \(a\) and \(b\), we can find their sum: \(a+b=(3x+2) + (3y+2) = 3(x+y)+4\).
03

Show that the sum is congruent to 1 modulo 3

Now we will show that \(a+b \equiv 1 (\bmod 3)\). Notice that we can rewrite the expression for the sum as: \(3(x+y)+4 = 3(x+y)+3+1 = 3(x+y+1)+1\). Since 3 divides \(3(x+y+1)\), we can say that \(a+b \equiv 1 (\bmod 3)\), which proves part (a).
04

Find the product using the given expressions for a and b

To prove that the product has a remainder of 1 when divided by 3, we need to show that \(a \cdot b \equiv 1 (\bmod 3)\). Using the expressions given above for \(a\) and \(b\), we can find their product: \(a \cdot b = (3x+2)(3y+2) = 9xy + 6x + 6y + 4\).
05

Show that the product is congruent to 1 modulo 3

Now we will show that \(a \cdot b \equiv 1 (\bmod 3)\). Notice that we can rewrite the expression for the product as: \(9xy + 6x + 6y + 4 = 9xy + 6(x+y) + 3+1 = 3(3xy + 2(x+y)+1) + 1\). Since 3 divides \(3(3xy + 2(x+y)+1)\), we can say that \(a \cdot b \equiv 1 (\bmod 3)\), which proves part (b).

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.

Mathematical Proof
In mathematics, a proof is a logical argument that establishes the truth of a statement. It is a sequence of steps that starts from agreed-upon axioms and applies rules of inference to arrive at a conclusion.

For example, in the exercise where we want to prove certain properties of modular congruence, the proof relies on clearly defined steps that build upon the definition of modular congruence and properties of integers. Specifically, the proof demonstrates how, given two numbers congruent to 2 modulo 3, their sum and product exhibit predictable behavior modulo 3.

Essentials of Mathematical Proof

In constructing a proof, it's essential to:
  • Start from known truths or previously established statements.
  • Use logical deductions to arrive at new conclusions.
  • Ensure that each step is justified and clearly explained.
  • End with the statement that you intended to prove.
By systematically following these steps, we validate our initial claim, providing a foundation of certainty on which mathematics depends.
Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers where numbers 'wrap around' after they reach a certain value鈥攖he modulus. Think of it like the face of a clock with the modulus being 12: 14 o'clock is the same as 2 o'clock because the numbers wrap around after reaching 12.

The notation used is the congruence symbol: if two numbers, say a and b, have the same remainder when divided by a modulus m, we can write, \(a \equiv b (\bmod m)\).

Applications of Modular Arithmetic

Modular arithmetic is not just an academic exercise鈥攊t's used in computer science for hashing algorithms, in cryptography for encryption, and even in everyday tasks like time calculation. Its periodic nature makes it excellent for anything that cycles or repeats.
Integer Properties
When working with integers in mathematics, we utilize certain properties to help us manipulate and understand them. Among these properties are:
  • Commutativity: The order in which we add or multiply integers does not affect the result (e.g., \(a + b = b + a\) and \(ab = ba\)).
  • Associativity: The way integers are grouped when adding or multiplying does not change the result (e.g., \(a + (b + c) = (a + b) + c\) and \(a(bc) = (ab)c\)).
  • Distributive property: Multiplication distributes over addition (e.g., \(a(b + c) = ab + ac\)).
In our exercise, these properties guide the transformation of expressions like \(a + b\) and \(a \cdot b\) into forms that reveal their modular relationships. This allows us to show that both expressions are congruent to 1 modulo 3.
Congruence Class
A congruence class is a set of numbers that share the same remainder when divided by a modulus. In simpler terms, it's like a club where the entrance requirement is 'having the same remainder.' For a given modulus m, the congruence class of a number n is all numbers that are congruent to n modulo m.

Two critical ideas concerning congruence classes are:
  • All members of a congruence class are interchangeable in modular equations. They 'represent' the same element in modular arithmetic.
  • Each integer belongs to exactly one congruence class modulo m.
In the context of our exercise, the numbers a and b both belong to the same congruence class modulo 3 (the '2 club'). This shared property is what allows us to predict and prove their behavior when summed and multiplied.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

One of the most famous unsolved problems in mathematics is a conjecture made by Christian Goldbach in a letter to Leonhard Euler in 1742. The conjecture made in this letter is now known as Goldbach's Conjecture. The conjecture is as follows: Every even integer greater than 2 can be expressed as the sum of two (not necessarily distinct) prime numbers. Currently, it is not known if this conjecture is true or false. (a) Write \(50,142,\) and 150 as a sum of two prime numbers. (b) Prove the following: If Goldbach's Conjecture is true, then every integer greater than 5 can be written as a sum of three prime numbers. (c) Prove the following: If Goldbach's Conjecture is true, then every odd integer greater than 7 can be written as a sum of three odd prime numbers.

Determine if each of the following propositions is true or false. Justify each conclusion. (a) For all integers \(a\) and \(b,\) if \(a b \equiv 0(\bmod 6),\) then \(a \equiv 0(\bmod 6)\) or \(b \equiv 0(\bmod 6)\) (b) For each integer \(a\), if \(a \equiv 2(\bmod 8),\) then \(a^{2} \equiv 4(\bmod 8)\). (c) For each integer \(a,\) if \(a^{2} \equiv 4(\bmod 8),\) then \(a \equiv 2(\bmod 8)\).

Prove that for all integers \(a\) and \(m,\) if \(a\) and \(m\) are the lengths of the sides of a right triangle and \(m+1\) is the length of the hypotenuse, then \(a\) is an odd integer.

Are the following propositions true or false? Justify all your conclusions. If a biconditional statement is found to be false, you should clearly determine if one of the conditional statements within it is true. In that case, you should state an appropriate theorem for this conditional statement and prove it. (a) For all integers \(m\) and \(n, m\) and \(n\) are consecutive integers if and only if 4 divides \(\left(m^{2}+n^{2}-1\right)\) (b) For all integers \(m\) and \(n, 4\) divides \(\left(m^{2}-n^{2}\right)\) if and only if \(m\) and \(n\) are both even or \(m\) and \(n\) are both odd.

Let \(h\) and \(k\) be real numbers and let \(r\) be a positive number. The equation for a circle whose center is at the point \((h, k)\) and whose radius is \(r\) is $$(x-h)^{2}+(y-k)^{2}=r^{2}$$ We also know that if \(a\) and \(b\) are real numbers, then The point \((a, b)\) is inside the circle if \((a-h)^{2}+(b-k)^{2}r^{2}\). Prove that all points on or inside the circle whose equation is \((x-1)^{2}+\) \((y-2)^{2}=4\) are inside the circle whose equation is \(x^{2}+y^{2}=26\).
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.