The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
The document discusses finding the least common multiple (LCM) and least common denominator (LCD) of numbers. It provides examples of:
1) Finding the LCM of numbers by listing their common multiples or constructing it from their prime factorizations.
2) Using the LCM, also called the LCD, of denominators to divide a pizza evenly between people who each want a fractional amount.
14 lcm, addition and subtraction of fractionsalg1testreview
The document discusses methods for finding the least common multiple (LCM) of two or more numbers. It defines the LCM as the smallest number that is a multiple of all the numbers. Two methods are described: listing the multiples and finding the smallest common one, and constructing the LCM by factorizing each number and taking the highest power of each prime factor. An example using each method is provided to find the LCM of different sets of numbers.
1 5 multiplication and division of rational expressionsmath123b
The document discusses methods for finding the least common multiple (LCM) of numbers. It describes the searching method, which involves finding the smallest number that is a multiple of all the given numbers. An example finds the LCM of 18, 24, 16 to be 144. The document also introduces the construction method, which builds the minimum coverage needed to fulfill all requirements.
The document discusses using the least common multiple (LCD) to convert fractions to equivalent whole numbers. It provides an example of finding the LCD of fractions {2/3, 5/8, 7/12, 3/4} which is 24. Multiplying each fraction by the LCD converts them to the whole number list {16, 15, 14, 18}. The fractions are then listed from largest to smallest as 3/4, 2/3, 5/8, 7/12.
14 the lcm and the multiplier method for addition and subtraction of rational...elem-alg-sample
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
2 the least common multiple and clearing the denominatorsmath123b
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required across different college applications in each subject area.
The document defines the least common multiple (LCM) as the smallest positive number that is a multiple of two or more given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from the prime factorizations. When the numbers are large, constructing the LCM is easier than listing multiples. The process of construction involves factorizing each number into prime factors and taking the highest power of each prime factor. The product of these highest powers gives the LCM. The LCM of denominators of fractions is also defined as the least common denominator (LCD).
3 lcm and lcd, addition and subtraction of fractionselem-alg-sample
The document defines the least common multiple (LCM) as the smallest positive number that is a multiple of two or more given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from prime factorizations. The constructing method involves factorizing each number, identifying the highest power of each prime factor, and taking their product. This is described as easier than listing when the LCM is large. The least common denominator (LCD) of fractions is also defined as the LCM of the denominators.
The document discusses finding the least common multiple (LCM) and least common denominator (LCD) of numbers. It provides examples of:
1) Finding the LCM of numbers by listing their common multiples or constructing it from their prime factorizations.
2) Using the LCM, also called the LCD, of denominators to divide a pizza evenly between people who each want a fractional amount.
14 lcm, addition and subtraction of fractionsalg1testreview
The document discusses methods for finding the least common multiple (LCM) of two or more numbers. It defines the LCM as the smallest number that is a multiple of all the numbers. Two methods are described: listing the multiples and finding the smallest common one, and constructing the LCM by factorizing each number and taking the highest power of each prime factor. An example using each method is provided to find the LCM of different sets of numbers.
1 5 multiplication and division of rational expressionsmath123b
The document discusses methods for finding the least common multiple (LCM) of numbers. It describes the searching method, which involves finding the smallest number that is a multiple of all the given numbers. An example finds the LCM of 18, 24, 16 to be 144. The document also introduces the construction method, which builds the minimum coverage needed to fulfill all requirements.
The document discusses using the least common multiple (LCD) to convert fractions to equivalent whole numbers. It provides an example of finding the LCD of fractions {2/3, 5/8, 7/12, 3/4} which is 24. Multiplying each fraction by the LCD converts them to the whole number list {16, 15, 14, 18}. The fractions are then listed from largest to smallest as 3/4, 2/3, 5/8, 7/12.
14 the lcm and the multiplier method for addition and subtraction of rational...elem-alg-sample
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
2 the least common multiple and clearing the denominatorsmath123b
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required across different college applications in each subject area.
The document defines the least common multiple (LCM) as the smallest positive number that is a multiple of two or more given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from the prime factorizations. When the numbers are large, constructing the LCM is easier than listing multiples. The process of construction involves factorizing each number into prime factors and taking the highest power of each prime factor. The product of these highest powers gives the LCM. The LCM of denominators of fractions is also defined as the least common denominator (LCD).
3 lcm and lcd, addition and subtraction of fractionselem-alg-sample
The document defines the least common multiple (LCM) as the smallest positive number that is a multiple of two or more given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from prime factorizations. The constructing method involves factorizing each number, identifying the highest power of each prime factor, and taking their product. This is described as easier than listing when the LCM is large. The least common denominator (LCD) of fractions is also defined as the LCM of the denominators.
The document discusses finding the least common multiple (LCM) of numbers. It defines the LCM as the smallest number that is a multiple of all the given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from prime factorizations. The preferred method when the LCM is large is to construct it by fully factorizing each number into prime factors and taking the highest power of each prime factor.
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required for each subject across different college requirements.
The document discusses the least common multiple (LCM) of numbers. The LCM of two or more numbers is the smallest positive number that is a multiple of each number. Two methods are presented for finding the LCM: 1) listing the multiples of each number and finding their common multiples, and 2) factorizing each number and taking the highest power of each prime factor. Examples are provided to illustrate both methods. The least common denominator (LCD) of fractions is also introduced as being the LCM of the denominators.
The document discusses methods for finding the least common multiple (LCM) of two or more numbers. It defines the LCM as the smallest number that is a multiple of all the numbers. Two methods are described: listing the multiples and finding their common multiples, and constructing the LCM by factorizing each number and taking the highest power of each prime factor. An example uses the constructing method to find the LCM of 8, 15, and 18 as 360.
This document discusses different methods for finding the least common multiple (LCM) of two or more numbers. It explains that the LCM is the smallest number that the given numbers will all divide into evenly. Then it describes four methods: multiplication, prime factorization, Venn diagrams, and using the greatest common factor (GCF). For each method it provides an example of finding the LCM of 24 and 18, and the answer is consistently 72.
Common Multiples and Least Common MultipleBrooke Young
The document explains how to find the least common multiple (LCM) of two numbers. It defines key terms like product, multiple, and common multiple. It then provides examples of finding the LCM of 3 and 6, 9 and 12, and 4 and 6. For each example, it lists the multiples of each number, circles the common multiples, and identifies the smallest common multiple as the LCM. The overall process is to list multiples, identify common multiples, and select the smallest value from the common multiples as the LCM.
The document provides instructions and examples for finding the least common multiple (LCM) for pairs of numbers. It lists 5 problems that ask the reader to identify the multiples of each number and determine the LCM. An answer key is provided that shows the multiples for each number and the calculated LCM for each pair.
The document discusses finding the least common multiple (LCM) of numbers. It provides examples of finding the LCM of pairs of numbers by listing their multiples and finding the smallest number that is a multiple of both. It also introduces finding the LCM using prime factorizations and Venn diagrams, showing how to find the greatest power of each prime factor and multiply them to get the LCM. Students are given practice problems to find the LCM of various number pairs using these methods.
The document discusses the concept of the least common multiple (LCM). It defines the LCM as the lowest number that is a multiple of two or more numbers. It provides examples of finding the LCM of different pairs of numbers by listing their multiples and circling the first number that is common to both lists. The document also discusses how the LCM can be used to find patterns involving multiples and to add or subtract fractions by finding a common denominator.
The document discusses finding the least common multiple (LCM) of two numbers. It defines a multiple as the result of multiplying a number by the counting numbers. The LCM is the smallest number that is common between the lists of multiples of two given numbers. Examples are provided to demonstrate finding the LCM by listing out the multiples of each number and identifying the first number that appears in both lists.
Finds the common multiples and the least common demo teachrosalio baybayan jr
The document outlines a lesson plan on finding common multiples and least common multiples (LCM) using different methods. It explains key terms like LCM, common, and multiple. Examples are given to demonstrate finding the LCM through listing common multiples, factor trees, or continuous division. Practice problems are provided for students to find the LCM of
This document contains a lesson on finding the least common multiple (LCM) and least common denominator (LCD) of numbers. It defines key terms like multiple, LCM, LCD and provides examples of using the list method and prime factorization to calculate the LCM and LCD of various number sets. Examples include finding the LCM of 6 and 4, the LCD of fractions with denominators of 4 and 2, and determining the minimum number of packs of hats and shirts needed for a boy scout troop.
This document provides an introduction to number theory, covering divisibility, greatest common divisors, least common multiples, and modular arithmetic. It defines key concepts such as division, factors, multiples, and the division algorithm. It presents theorems about divisibility and the relationship between greatest common divisors and least common multiples. Examples are provided to illustrate divisibility, the division algorithm, greatest common divisors, least common multiples, and modular arithmetic. The document serves as an overview of fundamental topics in number theory.
This document discusses determining common multiples and common factors of numbers. It explains that the common multiples of two numbers are the multiples that are shared between the two lists of all their individual multiples. The least common multiple is the smallest number that is a multiple of both numbers. It also explains that common factors are factors that two numbers have in common, and these can be determined by making lists of all the factors of each number and looking for the ones they share. A Venn diagram can also be used to visualize common factors between two numbers.
This document provides lessons and examples on ratio and proportion concepts to help students with analytical skills. It includes solved examples of ratio and proportion word problems. The document was created by Dr. T.K. Jain for free online entrepreneurship programs. It encourages students to help spread knowledge and social entrepreneurship. Links are provided to download additional free study materials on various topics.
The document provides examples and definitions for various math concepts including:
- Prime and composite numbers with examples of each
- Divisibility rules for numbers 3 through 10
- Prime factorization and examples factorizing 36, 54, and 99
- Definitions and examples of factors, greatest common factor (GCF), least common multiple (LCM), improper and mixed fractions, adding and subtracting fractions, multiplying and dividing fractions, ratios, proportions, percents and converting between percents and decimals, solving equations, and word problems involving fractions, ratios, and percents.
This document discusses lowest common multiples, factors, highest common factors, and prime numbers. It provides examples of finding the LCM and HCF of various number pairs. It defines a factor as a number that divides evenly into another number. Prime numbers are defined as numbers divisible only by 1 and themselves. Examples are given of listing the prime factors of different numbers.
This document appears to be a record of a student named Bryan Fabian Mejia Vergara completing an English workbook for the second partial exam of the 5th grade. It provides the student's name, subject, exam information and indicates the workbook was completed.
Respuestas diapositiva 13 factorizacion de polinomiosflor2510
Este documento contiene 8 expresiones algebraicas que involucran polinomios, raíces cuadradas y factores. Las expresiones incluyen términos como x2, x3, x5 y constates racionales multiplicados por factores polinómicos entre paréntesis.
53 multiplication and division of rational expressionsalg1testreview
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the tops divided by the product of the bottoms. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
This document discusses the importance of CEOs being marketing champions. It notes that CMOs only have an average tenure of 23 months. It then outlines various responsibilities and roles of effective marketing leaders, including aligning company capabilities with customer needs, injecting value before products launch, and driving business growth. The document emphasizes the need for marketing leaders to understand their strengths and focus, and lists attributes of those who drive superior revenue growth. It provides a checklist of metrics and indicators for marketing accountability and influence. Finally, it discusses key areas marketing should focus on, such as pricing, conversions, compelling events, and aligning product, channels, promotion and transactions to meet sales quotas.
The document discusses finding the least common multiple (LCM) of numbers. It defines the LCM as the smallest number that is a multiple of all the given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from prime factorizations. The preferred method when the LCM is large is to construct it by fully factorizing each number into prime factors and taking the highest power of each prime factor.
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required for each subject across different college requirements.
The document discusses the least common multiple (LCM) of numbers. The LCM of two or more numbers is the smallest positive number that is a multiple of each number. Two methods are presented for finding the LCM: 1) listing the multiples of each number and finding their common multiples, and 2) factorizing each number and taking the highest power of each prime factor. Examples are provided to illustrate both methods. The least common denominator (LCD) of fractions is also introduced as being the LCM of the denominators.
The document discusses methods for finding the least common multiple (LCM) of two or more numbers. It defines the LCM as the smallest number that is a multiple of all the numbers. Two methods are described: listing the multiples and finding their common multiples, and constructing the LCM by factorizing each number and taking the highest power of each prime factor. An example uses the constructing method to find the LCM of 8, 15, and 18 as 360.
This document discusses different methods for finding the least common multiple (LCM) of two or more numbers. It explains that the LCM is the smallest number that the given numbers will all divide into evenly. Then it describes four methods: multiplication, prime factorization, Venn diagrams, and using the greatest common factor (GCF). For each method it provides an example of finding the LCM of 24 and 18, and the answer is consistently 72.
Common Multiples and Least Common MultipleBrooke Young
The document explains how to find the least common multiple (LCM) of two numbers. It defines key terms like product, multiple, and common multiple. It then provides examples of finding the LCM of 3 and 6, 9 and 12, and 4 and 6. For each example, it lists the multiples of each number, circles the common multiples, and identifies the smallest common multiple as the LCM. The overall process is to list multiples, identify common multiples, and select the smallest value from the common multiples as the LCM.
The document provides instructions and examples for finding the least common multiple (LCM) for pairs of numbers. It lists 5 problems that ask the reader to identify the multiples of each number and determine the LCM. An answer key is provided that shows the multiples for each number and the calculated LCM for each pair.
The document discusses finding the least common multiple (LCM) of numbers. It provides examples of finding the LCM of pairs of numbers by listing their multiples and finding the smallest number that is a multiple of both. It also introduces finding the LCM using prime factorizations and Venn diagrams, showing how to find the greatest power of each prime factor and multiply them to get the LCM. Students are given practice problems to find the LCM of various number pairs using these methods.
The document discusses the concept of the least common multiple (LCM). It defines the LCM as the lowest number that is a multiple of two or more numbers. It provides examples of finding the LCM of different pairs of numbers by listing their multiples and circling the first number that is common to both lists. The document also discusses how the LCM can be used to find patterns involving multiples and to add or subtract fractions by finding a common denominator.
The document discusses finding the least common multiple (LCM) of two numbers. It defines a multiple as the result of multiplying a number by the counting numbers. The LCM is the smallest number that is common between the lists of multiples of two given numbers. Examples are provided to demonstrate finding the LCM by listing out the multiples of each number and identifying the first number that appears in both lists.
Finds the common multiples and the least common demo teachrosalio baybayan jr
The document outlines a lesson plan on finding common multiples and least common multiples (LCM) using different methods. It explains key terms like LCM, common, and multiple. Examples are given to demonstrate finding the LCM through listing common multiples, factor trees, or continuous division. Practice problems are provided for students to find the LCM of
This document contains a lesson on finding the least common multiple (LCM) and least common denominator (LCD) of numbers. It defines key terms like multiple, LCM, LCD and provides examples of using the list method and prime factorization to calculate the LCM and LCD of various number sets. Examples include finding the LCM of 6 and 4, the LCD of fractions with denominators of 4 and 2, and determining the minimum number of packs of hats and shirts needed for a boy scout troop.
This document provides an introduction to number theory, covering divisibility, greatest common divisors, least common multiples, and modular arithmetic. It defines key concepts such as division, factors, multiples, and the division algorithm. It presents theorems about divisibility and the relationship between greatest common divisors and least common multiples. Examples are provided to illustrate divisibility, the division algorithm, greatest common divisors, least common multiples, and modular arithmetic. The document serves as an overview of fundamental topics in number theory.
This document discusses determining common multiples and common factors of numbers. It explains that the common multiples of two numbers are the multiples that are shared between the two lists of all their individual multiples. The least common multiple is the smallest number that is a multiple of both numbers. It also explains that common factors are factors that two numbers have in common, and these can be determined by making lists of all the factors of each number and looking for the ones they share. A Venn diagram can also be used to visualize common factors between two numbers.
This document provides lessons and examples on ratio and proportion concepts to help students with analytical skills. It includes solved examples of ratio and proportion word problems. The document was created by Dr. T.K. Jain for free online entrepreneurship programs. It encourages students to help spread knowledge and social entrepreneurship. Links are provided to download additional free study materials on various topics.
The document provides examples and definitions for various math concepts including:
- Prime and composite numbers with examples of each
- Divisibility rules for numbers 3 through 10
- Prime factorization and examples factorizing 36, 54, and 99
- Definitions and examples of factors, greatest common factor (GCF), least common multiple (LCM), improper and mixed fractions, adding and subtracting fractions, multiplying and dividing fractions, ratios, proportions, percents and converting between percents and decimals, solving equations, and word problems involving fractions, ratios, and percents.
This document discusses lowest common multiples, factors, highest common factors, and prime numbers. It provides examples of finding the LCM and HCF of various number pairs. It defines a factor as a number that divides evenly into another number. Prime numbers are defined as numbers divisible only by 1 and themselves. Examples are given of listing the prime factors of different numbers.
This document appears to be a record of a student named Bryan Fabian Mejia Vergara completing an English workbook for the second partial exam of the 5th grade. It provides the student's name, subject, exam information and indicates the workbook was completed.
Respuestas diapositiva 13 factorizacion de polinomiosflor2510
Este documento contiene 8 expresiones algebraicas que involucran polinomios, raíces cuadradas y factores. Las expresiones incluyen términos como x2, x3, x5 y constates racionales multiplicados por factores polinómicos entre paréntesis.
53 multiplication and division of rational expressionsalg1testreview
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the tops divided by the product of the bottoms. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
This document discusses the importance of CEOs being marketing champions. It notes that CMOs only have an average tenure of 23 months. It then outlines various responsibilities and roles of effective marketing leaders, including aligning company capabilities with customer needs, injecting value before products launch, and driving business growth. The document emphasizes the need for marketing leaders to understand their strengths and focus, and lists attributes of those who drive superior revenue growth. It provides a checklist of metrics and indicators for marketing accountability and influence. Finally, it discusses key areas marketing should focus on, such as pricing, conversions, compelling events, and aligning product, channels, promotion and transactions to meet sales quotas.
El patrimonio cultural inmaterial del Perú incluye manifestaciones como arte, artesanía, danzas, bailes, festividades, gastronomía y música. El Perú cuenta con doce sitios considerados Patrimonio de la Humanidad por la UNESCO, incluyendo lugares en Cuzco, Nazca, Arequipa y Lima. La llegada de los conquistadores españoles introdujo la religión católica y produjo un sincretismo religioso a través de la evangelización. El Perú es uno de los países más diversos con seis biomas terrest
Ppt on karyotyping, chromosome banding and chromosome painting.ICRISAT
This document provides an overview of karyotyping, chromosome banding, and chromosome painting techniques. It discusses how karyotyping involves arranging chromosomes based on size and centromere position. Different banding techniques like Q, G, N, and C banding stain chromosomes to reveal structural features. Chromosome painting uses fluorescent probes to identify chromosomes and chromosomal abnormalities. These techniques allow studying chromosome structure, identifying defects, and analyzing evolutionary changes.
El documento describe la teoría del aprendizaje significativo de David Ausubel, la cual valora la experiencia del estudiante e identifica la importancia de relacionar conocimientos nuevos con los antiguos. También define el aprendizaje autónomo como asumir la responsabilidad de organizar el aprendizaje ajustándolo al propio tiempo, y enumera aptitudes como la motivación e interés que contribuyen a un aprendizaje significativo.
The document discusses methods for finding the least common multiple (LCM) of numbers. It describes the searching method, which involves finding the smallest number that is a multiple of all the given numbers. An example finds the LCM of 18, 24, 16 to be 144. The document also introduces the construction method, which builds the minimum coverage needed to fulfill all requirements.
The least common multiple (LCM) is the smallest number that is a multiple of both numbers. To find the LCM of two numbers:
1) List the multiples of each number
2) The LCM is the first number that appears in both lists
3) If no common multiple is found, continue listing multiples of each number until one is found
This document provides instruction on finding the least common multiple (LCM) of two or more numbers using different strategies. It explains that the LCM is the smallest number that is a multiple of both/all numbers. Three methods are described: 1) listing the multiples of each number and finding their common multiples, 2) using prime factorizations and factor trees, and 3) the Limon CAKE method which involves setting up the numbers in a cake-shape and multiplying the left and bottom parts. Examples of finding the LCM of various number pairs are provided and worked through step-by-step using the different methods.
This document provides instruction on finding the least common multiple (LCM) of two or more numbers using different strategies. It explains that the LCM is the smallest number that is a multiple of both/all numbers. Three methods are described: 1) listing the multiples of each number and finding their common multiples, 2) using prime factorization and factor trees, and 3) using the Lemon CAKE method which involves setting up a division problem to systematically find the LCM. Examples of each method are worked through.
This document discusses factors, multiples, least common multiples (LCM), and greatest common factors (GCF). It provides examples of finding the multiples of numbers by multiplying them by counting numbers. It also shows how to find the LCM and GCF of two numbers by listing their common multiples or factors and taking the least or greatest. The key steps are to find the multiples of each number, identify their common multiples to determine the LCM, or common factors to determine the GCF.
This document discusses finding the least common multiple (LCM) of sets of numbers. It begins with examples of finding the LCM of various number pairs and sets. These examples illustrate finding the prime factors of each number and identifying the smallest number that is a multiple of all numbers as their LCM. The document then provides practice problems for readers to identify the LCM of additional number sets. It concludes by recapping that the LCM is the smallest number that is divisible by all numbers in the set.
The document introduces the concept of the least common multiple (LCM) and provides examples of finding the LCM of numbers using their prime factorizations or a Venn diagram. It explains that the LCM is the smallest number that is a multiple of both numbers. Students are given practice problems to find the LCM of number pairs and expressions using these methods.
1) Logarithms provide an alternative way to express exponential expressions like 16=24 by writing log216=4, where the logarithm is equivalent to the power or index in the original expression.
2) The three laws of logarithms describe relationships between logarithms and exponents: the first law states that loga(xy)=logax + logay, the second law states that loga(xm)= mlogax, and the third law states that loga(x/y)=logax - logay.
3) Logarithms can be used to solve equations where the unknown is in the power by taking logarithms of both sides and using the laws of logarithms to isolate the unknown.
What is HCF? What is LCM? How you calculate HCF & LCM
of numbers & fractions quickly?
Find out in this short presentation by https://allexammocktest.in
The document discusses greatest common factor (GCF) and least common multiple (LCM). It provides examples of factors and multiples, and describes three methods for finding the GCF and LCM of two numbers: listing factors/multiples, prime factorization, and using division. The document also discusses how to identify if a word problem requires using the GCF or LCM to solve, and provides an example word problem for each.
The document discusses properties of rhombi and squares. It defines a rhombus as a parallelogram with four congruent sides and gives its properties. It defines a square as a parallelogram with four right angles and four congruent sides, having properties of rectangles, rhombi, and parallelograms. Theorems are presented about diagonals of rhombi and conditions for rhombi and squares. Examples solve problems about finding angle measures in rhombi and proving a parallelogram is a rhombus.
The document discusses highest common factors (HCF) and lowest common multiples (LCM) of numbers. It provides examples of finding the HCF and LCM of different sets of numbers. The HCF is the largest number that divides evenly into all numbers. The LCM is the smallest number that all numbers divide evenly into. The document also discusses prime numbers and factoring numbers to find their HCF and LCM.
This document discusses highest common factors (HCF) and least common multiples (LCM). It provides examples of calculating HCF and LCM using prime factorization and division methods for numbers, polynomials, and fractions. The key points are:
- HCF is the greatest number that divides two or more numbers. LCM is the smallest number divisible by two or more numbers.
- Prime factorization and division methods can be used to calculate HCF and LCM of numbers.
- For polynomials, the product of common factors is the HCF and the product of factors with highest powers is the LCM.
- For fractions, the HCF is the HCF of numerators and LCM of denomin
This document provides an introduction to number theory, covering divisibility, greatest common divisors, least common multiples, and modular arithmetic. It defines key concepts such as division, factors, multiples, and the division algorithm. It presents theorems about divisibility and the relationship between greatest common divisors and least common multiples. Examples are provided to illustrate divisibility, the division algorithm, greatest common divisors, least common multiples, and modular arithmetic. The document serves as an overview of fundamental topics in number theory.
This document provides examples and explanations for solving linear equations with one variable. It begins by defining algebraic expressions, variables, and algebraic equations. It then focuses on linear equations in one variable where the variable has an exponent of 1. Examples are provided for solving linear equations by transposing terms to one side of the equation and using addition or subtraction to simplify. Word problems involving linear equations are also worked through. Finally, techniques for solving linear equations with fractions or rational numbers are described. The key steps covered are writing an equation from a word problem, transposing terms, adding/subtracting to isolate the variable, and solving for the variable value.
This document discusses prime factorization, greatest common factors (GCF), least common multiples (LCM), and the Euclidean algorithm. It provides examples of finding the prime factorization of numbers, using factor trees and canonical representation. It also gives steps for calculating the GCF and LCM of two or more numbers by finding their prime factorizations and examining the exponents of common factors. The Euclidean algorithm is introduced as an alternate method for finding the GCF of two numbers based on repeated division. Applications of these concepts to rational numbers and fractions are also described.
A prime number is a whole number greater than 1 that is only divisible by 1 and itself. Examples of prime numbers are 2, 3, 5. A composite number is a whole number that has more than two factors. Examples of composite numbers are 4, 6, 9.
The divisibility rules for 3, 4, 6, 9, 10 are: a number is divisible by 3 if the sum of its digits is divisible by 3; a number is divisible by 4 if the last two digits are divisible by 4; a number is divisible by 6 if it is divisible by both 2 and 3; a number is divisible by 9 if the sum of its digits is divisible by 9; a number is divisible by 10 if
The document discusses systems of linear equations. It begins by explaining that to solve for one unknown quantity, one piece of information is needed, and to solve for two unknowns, two pieces of information are needed. This leads to systems of linear equations, which are collections of two or more linear equations with two or more variables. A solution to a system is a set of numbers for the variables that satisfies all equations. An example system is provided. The document then works through an example problem about the cost of hamburgers and salads to demonstrate solving a system of linear equations.
55 addition and subtraction of rational expressions alg1testreview
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding and subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate adding and subtracting rational expressions, as well as converting them to have a common denominator.
The document discusses rational expressions, which are expressions of the form P/Q where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They can be written in expanded or factored form. The factored form is useful for solving equations, determining the domain of valid inputs, evaluating expressions, and determining the sign of outputs. The domain excludes values that would make the denominator equal to 0. Solutions to equations involving rational expressions are the zeros of the numerator polynomial P.
The document discusses basic geometrical shapes and formulas for calculating their perimeters. It defines a loop or polygon as a shape formed when the ends of a rope or string resting in a plane are connected. Plane shapes enclosed by straight lines are called polygons, including triangles, rectangles, and squares. Formulas are provided for calculating the perimeters of triangles, rectangles, squares, and combinations of shapes. Examples are included to demonstrate calculating total perimeter lengths required for fencing or roping off areas.
The document discusses mathematical expressions and how to combine them. It defines expressions as calculation procedures written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms are combined by adding or subtracting coefficients in the same way numbers are combined. Unlike terms, such as x-terms and number terms, cannot be combined.
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined, and tilted lines. Horizontal lines have the equation y=c, vertical lines x=c, and tilted lines are found using the point-slope formula y=m(x-x1)+y1, where m is the slope and (x1,y1) is a point on the line. Examples are given to demonstrate finding equations of lines given information about them.
The document discusses linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
The document defines slope as the ratio of the rise (change in y-values) to the run (change in x-values) between two points on a line. It provides the exact formula for calculating slope as the change in y-values divided by the change in x-values. Examples are given to demonstrate calculating the slopes of various lines, with positive slopes for lines passing through Quadrants I and III and negative slopes for lines passing through Quadrants II and IV.
55 inequalities and comparative statementsalg1testreview
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. This line is called the real number line. Two numbers are related by an inequality if their corresponding positions on the real number line have one number being further to the right than the other. Inequalities can be used to represent intervals of numbers on the real number line.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance right or left from the origin and y is the distance up or down. This divides the plane into four quadrants, with the signs of x and y determining which quadrant a point falls into. Examples are given of labeling points and finding coordinates on the grid.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to points to the right of 0, and negative numbers to points on the left. The number line defines the relative size of numbers based on their position, with numbers to the right being greater than those to the left. Intervals on the number line, such as -1 < x < 3, represent all the numbers between and including the bounds.
The document discusses the Pythagorean theorem and square roots. It begins by defining a right triangle and its components. It then states the Pythagorean theorem, which relates the lengths of the sides of a right triangle. An example problem demonstrates using the theorem to find the height of a wall. The document concludes by defining the square root, explaining that the square root of a number is its positive root and how to calculate and approximate square roots.
The document discusses basic facts about triangles, including:
- A triangle is a three-sided polygon formed by three line segments.
- The three angles of any triangle always sum to 180 degrees.
- The triangle inequality states that the sum of any two side lengths must be greater than the third side length.
- Two triangles are congruent if they have the same shape and size.
The document discusses solving linear equations by factoring using an example of determining the number of pizzas ordered. It formulates the problem as the equation 3x + 10 = 34 where x is the number of pizzas. It solves the equation by subtracting 10 from both sides, dividing both sides by 3, and determining that x = 8 pizzas were ordered. The document then provides more details on linear equations, their structure, and their general solution method.
The document discusses proportions and ratios. It defines a ratio as two related quantities stated side by side. It provides an example of a recipe ratio of 3 eggs to 4 cups of flour as 3:4. It explains how to set up proportional equations from word problems by ensuring quantities of the same type occupy the same position in fractions. It solves examples, finding x eggs needed given 10 cups of flour is 7.5 eggs. It also solves a map ratio problem of 4 inches on a map equaling 21 miles in real distance.
49 factoring trinomials the ac method and making listsalg1testreview
The document discusses factoring trinomials. It explains that trinomials are polynomials of the form ax2 + bx + c. There are two types of trinomials: those that are factorable, which can be written as the product of two binomials, and those that are not factorable. The ac-method uses tables to determine whether a trinomial is factorable or not by finding values for u and v that fit the table. If values are found, the trinomial can be factored using the grouping method. An example factors the trinomial x2 - x - 6 by grouping.
48 factoring out the gcf and the grouping methodalg1testreview
The document discusses factoring quantities and finding common factors and greatest common factors (GCF). It provides examples of factoring numbers like 12 and finding the common factors and GCF of expressions. Factoring means rewriting a quantity as a product without 1. A common factor is one that belongs to all quantities, and the GCF is the largest common factor considering coefficients and degrees of factors. Examples show finding the common factors of expressions and the GCF of quantities like 24 and 36.
47 operations of 2nd degree expressions and formulasalg1testreview
The document discusses operations involving binomials and trinomials. It defines a binomial as a two-term polynomial of the form ax + b and a trinomial as a three-term polynomial of the form ax2 + bx + c. It states that the product of two binomials is a trinomial that can be found using the FOIL method: multiplying the first, outer, inner, and last terms. The FOIL method is demonstrated through examples multiplying binomial expressions. Expanding products involving negative binomials requires distributing the negative sign before using FOIL.
The document discusses scientific notation, which is a way to write very large or small numbers in a standardized form. It explains that in scientific notation, any number can be written as A x 10N, where 1 < A < 10 and N is an exponent that represents powers of 10. Positive exponents mean the decimal is moved to the right, and negative exponents mean it is moved to the left. The document provides examples of writing numbers in scientific notation and changing them back to standard form by moving the decimal according to the exponent N.
The document discusses polynomial expressions. A polynomial is the sum of monomial terms, where a monomial is a number multiplied by one or more variables raised to a non-negative integer power. Examples show evaluating polynomials by substituting values for variables and calculating each monomial term separately before combining them. A term refers to each monomial within a polynomial. Terms are identified by their variable part, such as the x2-term, x-term, or constant term.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
2. The Least Common Multiple (LCM)
We say m is a multiple of x if m can be divided by x.
3. For example, 12 is a multiple of 2
The Least Common Multiple (LCM)
We say m is a multiple of x if m can be divided by x.
4. For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
We say m is a multiple of x if m can be divided by x.
5. For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
xy2 is a multiple of x,
We say m is a multiple of x if m can be divided by x.
6. For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
We say m is a multiple of x if m can be divided by x.
7. For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
We say m is a multiple of x if m can be divided by x.
8. For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
We say m is a multiple of x if m can be divided by x.
9. For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
10. The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
11. The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
12. The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6,
xy is the LCM of x and y,
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
13. The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6,
xy is the LCM of x and y,
and that 12xy is the LCM of 4x and 6y.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
14. The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6,
xy is the LCM of x and y,
and that 12xy is the LCM of 4x and 6y.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
We give two methods for finding the LCM.
I. The searching method
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
15. The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6,
xy is the LCM of x and y,
and that 12xy is the LCM of 4x and 6y.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
We give two methods for finding the LCM.
I. The searching method
II. The construction method
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
17. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Methods of Finding LCM
18. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
Methods of Finding LCM
19. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24:
Methods of Finding LCM
20. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72,
Methods of Finding LCM
21. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
Methods of Finding LCM
22. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
23. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Construction Method:
24. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Construction Method: Given two or more quantities,
I. factor each quantity completely,
25. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
26. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
27. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Factor each number completely
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
28. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Factor each number completely
18 = 2*32
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
29. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Factor each number completely
18 = 2*32
24 = 233
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
30. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Factor each number completely
18 = 2*32
24 = 233
20 = 225
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
31. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Factor each number completely
18 = 2*32
24 = 233
20 = 225
Take the highest of the each factor,
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
32. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Factor each number completely
18 = 2*32
24 = 233
20 = 225
Take the highest of the each factor,
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
33. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Factor each number completely
18 = 2*32
24 = 233
20 = 225
Take the highest of the each factor,
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
34. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Factor each number completely
18 = 2*32
24 = 233
20 = 225
Take the highest of the each factor,
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
35. Searching Method
Given a list of quantities, list the multiples of the largest
quantity and test them one by one until we find the LCM.
Example A. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
Example B. a. Construct the LCM of 18, 24, 20.
Factor each number completely
18 = 2*32
24 = 233
20 = 225
Take the highest of the each factor,
so the LCM is 23325 = 360.
Construction Method: Given two or more quantities,
I. factor each quantity completely,
II. multiply the highest power of each factor that appears in
the factorization to get the LCM.
36. b. Construct the LCM of x2y3z, x3yz4, x3zw
Methods of Finding LCM
37. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor.
Methods of Finding LCM
38. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM =
Methods of Finding LCM
39. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3
Methods of Finding LCM
40. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3
Methods of Finding LCM
41. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4
Methods of Finding LCM
42. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
43. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 – x – 2
44. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 – x – 2
Factor each quantity.
45. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
46. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
47. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM =
48. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)
49. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)
50. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
51. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
52. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
53. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4)
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
54. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
55. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
56. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
Take the highest power of each factor to get the
LCM =
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
57. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
Take the highest power of each factor to get the
LCM = x2
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
58. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
Take the highest power of each factor to get the
LCM = x2(x – 2)2
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
59. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, Take the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
Take the highest power of each factor to get the
LCM = x2(x – 2)2(x + 2)
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Take the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
60. The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list.
Methods of Finding LCM
61. The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method .
Methods of Finding LCM
62. Methods of Finding LCM
The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method .
Example C. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college, how
many years of each subject does she need?
63. Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Methods of Finding LCM
The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method .
Example C. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college, how
many years of each subject does she need?
64. Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Methods of Finding LCM
The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method .
Example C. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college, how
many years of each subject does she need?
65. Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Science-3 yr Math-3yrs English-4 yrs Arts-1yr
Methods of Finding LCM
The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method .
Example C. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college, how
many years of each subject does she need?
66. Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Science-3 yr Math-3yrs English-4 yrs Arts-1yr
Methods of Finding LCM
The Construction Method takes just enough of each factor
to make the LCM which "covers" all the quantities on the
list. The following is an example that illustrates the
principle behind this method .
Example C. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college, how
many years of each subject does she need?
So we take the highest power of each factor to make the LCM.