/*! 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 22 Show that \(\operatorname{lcm}(a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 22

Show that \(\operatorname{lcm}(a, b)=a b\) if and only if \(a\) and \(b\) are relatively prime.

Short Answer

Expert verified
\(a\) and \(b\) are relatively prime if and only if \(\operatorname{lcm}(a, b) = ab\).

Step by step solution

01

Define relatively prime

Two integers \(a\) and \(b\) are relatively prime if their greatest common divisor (\(\gcd\)) is 1, i.e. \(\gcd(a, b)=1\).
02

Understand the relationship between lcm and gcd

The least common multiple (\(\operatorname{lcm}(a, b)\)) of \(a\) and \(b\) is related to their gcd by the formula: \[\operatorname{lcm}(a, b) = \frac{a \times b}{\gcd(a, b)}.\]
03

Prove the 'if' part

If \(a\) and \(b\) are relatively prime, \(\gcd(a, b) = 1\). Substituting in the lcm-gcd relationship gives: \[\operatorname{lcm}(a, b) = \frac{a \times b}{1} = a \times b.\] This proves that if \(a\) and \(b\) are relatively prime, then \(\operatorname{lcm}(a, b) = a \times b\).
04

Prove the 'only if' part

Assume \(\operatorname{lcm}(a, b) = a \times b\). Using the relationship \(\operatorname{lcm}(a, b) = \frac{a \times b}{\gcd(a, b)}\), we have: \[a \times b = \frac{a \times b}{\gcd(a, b)}.\] Simplifying, \[\gcd(a, b) = 1.\] This means \(a\) and \(b\) are relatively prime.

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.

Relatively Prime Integers
Two numbers are considered relatively prime if they have no common factors other than 1. In mathematical terms, this means that their greatest common divisor (GCD) is precisely 1. For example, the numbers 8 and 15 are relatively prime because the only positive integer that divides both of them is 1. No other common factors exist between the two.

When numbers are relatively prime, it opens up interesting properties, especially in number theory. For instance:
  • When two numbers are relatively prime, their least common multiple (LCM) is simply the product of the numbers themselves.
  • When a number has a factor that is also a factor of another number, they cannot be relatively prime.
Exploring relationships between relatively prime numbers is crucial in understanding LCM and GCD relationships.
Greatest Common Divisor
The greatest common divisor (GCD) of two numbers is the largest integer that divides both numbers without leaving a remainder. An example is finding the GCD of 56 and 98, which is 14, since it's the biggest number that can evenly divide both 56 and 98.

Key points about the GCD include:
  • If the GCD of two numbers is 1, those numbers are relatively prime.
  • It plays a critical role in simplifying fractions, as any fraction can be reduced by dividing both numerator and denominator by their GCD.
  • The concept of GCD is also foundational in exploring mathematical concepts like LCM and their intertwined relationships.
Understanding GCD is essential to grasp the link between it and other properties like LCM.
Least Common Multiple
The least common multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers. For example, the LCM of 12 and 15 is 60 because 60 is the smallest number that both 12 and 15 can divide evenly into.

Some important points about LCM include:
  • If two numbers are relatively prime, the LCM is simply the product of those two numbers.
  • LCM is especially useful in solving problems where common denominators are needed, such as adding two fractions with different denominators.
  • The relationship between LCM and GCD is beautifully expressed by the equation: \(\operatorname{lcm}(a, b) = \frac{a \times b}{\gcd(a, b)}\). This equation clearly shows how LCM depends on both the multiplication of the numbers and their GCD.
Understanding LCM helps in solving practical arithmetic problems and grasping deeper mathematical principles.

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

$$ \begin{aligned} &\text { Show that } n \text { is prime if and only if in } \mathbb{Z}_{n},[r][s]=[0] \text { always implies }[r]=[0] \text { or }\\\ &[s]=[0] \end{aligned} $$

Determine whether the indicated relation is an equivalence relation on the indicated set and, if so, describe the equivalence classes. In \(\mathbb{R} a \sim b\) if and only if \(|a|=|b|\)

Calculate the determinant of the indicated matrix. $$ \left[\begin{array}{ll} 5 & 1 \\ 2 & 2 \end{array}\right] \text { in } \mathbb{Z}_{7} $$

Determine whether the indicated matrix is invertible and. if so, calculate the inverse matrix. $$ \left[\begin{array}{cc} 4 & 1 \\ -3 & 2 \end{array}\right] \text { in } M\left(2, Z_{5}\right) $$

Determine whether or not the indicated maps are invertible. $$ \phi: \mathbb{R} \rightarrow \mathbb{R}, \text { where } \phi(x)=(5 x+3) / 2 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.