Mathematics
Common Multiples
Common multiples are the multiples that two or more numbers have in common. In other words, they are the numbers that are multiples of each of the given numbers. Finding common multiples is useful in various mathematical operations, such as finding a common denominator for fractions or determining the least common multiple (LCM) of a set of numbers.
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4 Key excerpts on "Common Multiples"
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
• The lowest common multiple ( LCM ) of two or more numbers is the smallest possible number into which all the numbers will divide exactly. • Factorising Taking out the highest common factor. Common factors are the factors shared by two or more numbers. A common factor can be a number or a variable, or a combination of numbers and variables, for example: 4 x 2 y 3 z 2 + 16 x 3 y 4 z 2 – 64 xyz 3 = 4 xyz 2 ( xy 2 + 4 x 2 y 3 – 16 z ) Grouping An expression with more than three terms can be factorised by grouping terms to find a common factor, for example: ax + ay – bx – by = ( ax + ay ) + (– bx – by ) = a ( x + y ) + b (– x – y ) = a ( x + y ) – b ( x + y ) = ( x + y )( a + b ) • Algebraic fractions Rules of fractions Rule Explanation 1 a b × c d = ac bd Also: a × c d = a 1 × c d = ac d When multiplying fractions, multiply all the numerators and then all the denominators . Restrictions: b ≠ 0; d ≠ 0 105 Introductory Mathematics| Hands-On Rule Explanation 2 a b ÷ c d = a b × d c = ad bc When dividing fractions, change the operation from division to multiplication and invert whatever is after the division sign (multiplying with the reciprocal of the fraction). This is called the tips and times rule. Restrictions: b ≠ 0; c ≠ 0; d ≠ 0 3 a b + c b = a c b + or a b + c d = ad bc bd + To add or subtract fractions, the denominator must be the same . Restrictions: b ≠ 0; d ≠ 0 • Multiplying and dividing fractions Steps for multiplying fractions Step 1 Cancel factors in the numerator with factors in the denominator. Step 2 Multiply the numerators by numerators. Step 3 Multiply the denominators by denominators. Step 4 Simplify the answer to its simplest form. Steps for dividing fractions Step 1 First change the ÷ (division) sign to a × (multiplication) sign, and invert the fraction after the division sign. Step 2 Follow the steps above for multiplying fractions.
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Many of these topics are useful in other areas of mathematics such as fractions (Chapter 6) and algebra. Being able to find common factors and greatest common factors between pairs of numbers is useful when simplifying fractions. Understanding least Common Multiples is useful for finding a common denominator when adding and subtracting fractions. The principles of factor- ing numbers studied in this chapter generalize into similar situations in algebra when factoring expressions. Key Concepts from the NCTM Principles and Standards for School Mathematics PRE-K-2–NUMBER AND OPERATIONS Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers. GRADES 3-5–NUMBER AND OPERATIONS Recognize equivalent representations for the same number and generate them by decomposing and composing num- bers. Describe classes of numbers according to characteristics such as the nature of their factors. GRADES 6-8–NUMBER AND OPERATIONS Use factors, multiples, prime factorization, and relatively prime numbers to solve problems. Key Concepts from the NCTM Curriculum Focal Points GRADE 3: Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts. GRADE 4: Developing quick recall of multiplication facts and related division facts and fluency with whole-number multiplication. Key Concepts from the Common Core State Standards for Mathematics GRADE 4: Gain familiarity with factors and multiples. GRADE 6: Find common factors and multiples. Apply and extend previous understandings of arithmetic to alge- braic expressions. Section 5.1 Primes, Composites, and Tests for Divisibility 177 For example, 2, 3, 5, 7, 11 are primes, since they have only themselves and 1 as fac- tors; 4, 6, 8, 9, 10 are composites, since they each have more than two factors; 1 is neither prime nor composite, since 1 is its only factor.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Least Common Multiple and Euclidean Algorithm Least common multiple In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple ) of two integers a and b , usually denoted by LCM( a , b ) , is the smallest positive integer that is a multiple of both a and b . It is familiar from grade-school arithmetic as the lowest common denominator that must be determined before two fractions can be added. This definition may be extended to rational numbers a and b : the LCM is the smallest positive rational number that is an integer multiple of both a and b . (In fact, the definition may be extended to any two real numbers whose ratio is a rational number.) If either a or b is 0, LCM( a , b ) is defined to be zero. The LCM of more than two integers or rational numbers is well-defined: it is the smallest number that is an integer multiple of each of them. Examples Integer What is the LCM of 4 and 6? Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, ... and the multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, ... Common Multiples of 4 and 6 are simply the numbers that are in both lists: ________________________ WORLD TECHNOLOGIES ________________________ 12, 24, 36, 48, 60, .... So the least common multiple of 4 and 6 is the smallest one of those: 12 = 3 × 4 = 2 × 6. Rational What is the LCM of and The multiples of are: and the multiples of are: Therefore, their LCM is the smallest number on both lists. What is the LCM of and The multiples of are: and the multiples of are: So their LCM is Note that, by definition, if a and b are two rationals (or integers), there are integers m and n such that LCM( a , b ) = m × a = n × b .
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Euclidean geometry is also named after Euclid. Classroom Tip Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 184 Chapter 5 Number Theory The Least Common Multiple Definition of Least Common Multiple (LCM) The least common multiple (LCM) of two or more natural numbers is the least natural number that is a multiple of both or all of the natural numbers. The notation for the least common multiple of a and b is written as LCM(a, b). To find the LCM(a, b), write the prime factorizations of a and b using two intersecting sets. The least common multiple is the product of the factors in all three regions. a = 20 b = 12 2 2 5 3 Common factors Example a = 20, b = 12 a = 2 ⋅ 2 ⋅ 5, b = 2 ⋅ 2 ⋅ 3 LCM(20, 12) = 2 ⋅ 2 ⋅ 3 ⋅ 5 = 60 If the only common factor of a and b is 1, then the LCM(a, b) = a ⋅ b. Example LCM(3, 5) = 3 ⋅ 5 = 15 EXAMPLE 5 Finding the Least Common Multiple Find the least common multiple of each pair of whole numbers. a. 6, 10 b. 18, 20 SOLUTION a. Begin by writing the prime factorization of each number. a = 6: 6 b = 10: 10 2 3 2 5 So, 6 = 2 ⋅ 3 and 10 = 2 ⋅ 5. Next, use a diagram to organize the prime factors of a and b. The least common multiple of 6 and 10 is 2 ⋅ 3 ⋅ 5 = 30. b. Begin by writing the prime factorization of each number. a = 18: 18 b = 20: 20 2 9 4 5 3 3 2 2 So, 18 = 2 ⋅ 3 ⋅ 3 and 20 = 2 ⋅ 2 ⋅ 5. Next, use a diagram to organize the prime factors of a and b. The least common multiple of 18 and 20 is 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 180. The least common multiple of 1 and any nonzero whole number is simply the whole number.
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