/*! 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 31 Show that if \(a\) and \(b\) are... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 31

Show that if \(a\) and \(b\) are positive integers, then \(a b=\) \(\operatorname{gcd}(a, b) \cdot \operatorname{lcm}(a, b) .\) [Hint: Use the prime factorizations of \(a\) and \(b\) and the formulae for \(\operatorname{gcd}(a, b)\) and \(\operatorname{lcm}(a, b)\) in terms of these factorizations.

Short Answer

Expert verified
The product of gcd(a, b) and lcm(a, b) equals ab.

Step by step solution

01

Prime Factorization

Express the numbers a and b in their prime factorized forms. Let鈥檚 represent them as follows: a = p1^{e1} * p2^{e2} * ... * pn^{en} b = p1^{f1} * p2^{f2} * ... * pn^{fn} where pi represents a prime number, and ei and fi are their respective powers in the factorization of a and b.
02

Understanding gcd(a, b)

The greatest common divisor (gcd) of a and b is calculated by taking the lowest power of each prime that appears in both a and b: gcd(a, b) = p1^{min(e1, f1)} * p2^{min(e2, f2)} * ... * pn^{min(en, fn)}.
03

Understanding lcm(a, b)

The least common multiple (lcm) of a and b is found by taking the highest power of each prime that appears in either a or b: lcm(a, b) = p1^{max(e1, f1)} * p2^{max(e2, f2)} * ... * pn^{max(en, fn)}.
04

Multiply gcd(a, b) and lcm(a, b)

Multiply gcd(a, b) and lcm(a, b) together: gcd(a, b) * lcm(a, b) = (p1^{min(e1, f1)} * p2^{min(e2, f2)} * ... * pn^{min(en, fn)}) * (p1^{max(e1, f1)} * p2^{max(e2, f2)} * ... * pn^{max(en, fn)}).
05

Simplifying the Expression

Simplify the multiplication by combining the powers of the corresponding prime factors: (p1^{min(e1, f1) + max(e1, f1)} * p2^{min(e2, f2) + max(e2, f2)} * ... * pn^{min(en, fn) + max(en, fn)}). Notice that min(ei, fi) + max(ei, fi) = ei + fi for each i.
06

Result

Thus, combining the simplified terms, we have: gcd(a, b) * lcm(a, b) = p1^{e1 + f1} * p2^{e2 + f2} * ... * pn^{en + fn} = a * 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.

greatest common divisor
The greatest common divisor (gcd) of two positive integers, a and b, is the largest positive integer that divides both a and b without leaving a remainder. For any two numbers, their gcd can be calculated using the prime factorization method.

Here鈥檚 how you can compute it:
  • First, express both numbers in their prime factorized forms. For example, let's say \( a = p1^{e1} * p2^{e2} * ... * pn^{en} \) and \( b = p1^{f1} * p2^{f2} * ... * pn^{fn} \), where \( pi \) represents a prime number, and \( ei \) and \( fi \) are their respective powers in the factorization of a and b.
  • Next, identify the common prime factors and select the lowest power for each prime that appears in both numbers. The formula for gcd(a, b) in terms of their factorizations is: \[ \operatorname{gcd}(a, b) = p1^{\min(e1, f1)} * p2^{\min(e2, f2)} * ... * pn^{\min(en, fn)} \]
  • For example, for \( a = 24 \) and \( b = 36 \) the prime factorizations are \( 24 = 2^3 * 3^1 \) and \( 36 = 2^2 * 3^2 \). The gcd would be \( 2^{\min(3,2)} * 3^{\min(1,2)} = 2^2 * 3^1 = 12 \).
Understanding how to compute the gcd is essential because it is foundational for solving many problems involving number theory.
least common multiple
The least common multiple (lcm) of two positive integers, a and b, is the smallest positive integer that is divisible by both a and b. Calculating the lcm can also be done using the prime factorization method.

Here's a simple guide to find the lcm:
  • First, express both numbers in their prime factorized forms. Let鈥檚 represent them as: \( a = p1^{e1} * p2^{e2} * ... * pn^{en} \) and \( b = p1^{f1} * p2^{f2} * ... * pn^{fn} \).
  • Next, take each prime number that appears in either of the factorizations and choose the highest power for each prime that appears. The formula for the lcm is: \[ \operatorname{lcm}(a, b) = p1^{\max(e1, f1)} * p2^{\max(e2, f2)} * ... * pn^{\max(en, fn)} \]
  • For instance, for \( a = 24 \) and \( b = 36 \), the prime factorizations are \( 24 = 2^3 * 3^1 \) and \( 36 = 2^2 * 3^2 \). The lcm would be \( 2^{\max(3,2)} * 3^{\max(1,2)} = 2^3 * 3^2 = 72 \).
Mastering lcm is useful in solving various problems related to multiples and distributions.
prime factorization
Prime factorization is the process of expressing a positive integer as a product of prime numbers. This method is foundational in number theory. Here is how you can perform prime factorization:
  • Start by dividing the number by the smallest prime number (usually 2) and continue dividing by that prime as long as the division is exact (no remainder).
  • Once the division by the smallest prime is no longer possible, move to the next smallest prime number and repeat.
  • Continue this process until you're left with a quotient of 1.
For example, let鈥檚 factorize the number 60:
  • 60 梅 2 = 30
  • 30 梅 2 = 15
  • 15 梅 3 = 5 (note: 3 is the next smallest prime after 2)
  • 5 梅 5 = 1
So, the prime factorization of 60 is \( 2^2 * 3^1 * 5^1 \). Prime factorization helps in simplifying the computation of both gcd and lcm. It鈥檚 also valuable in studying and understanding the structure of numbers.

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

Show that a positive integer is divisible by 3 if and only if the sum of its decimal digits is divisible by \(3 .\)

Devise an algorithm that, given the binary expansions of the integers \(a\) and \(b\) , determines whether \(a>b, a=b,\) of \(a

Prove Euler's criterion, which states that if \(p\) is an odd prime and \(a\) is a positive integer not divisible by \(p\) , then $$ \left(\frac{a}{p}\right) \equiv a^{(p-1) / 2}(\bmod p) $$

The first nine digits of the ISBN-10 of the European version of the fifth edition of this book are 0-07-119881. What is the check digit for that book?

Give a simple algorithm for forming the two's complement representation of an integer from its one's complement representation.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.