5 Key excerpts on "Common Factors"

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  eBook - ePub

  Barron's Math 360: A Complete Study Guide to Pre-Algebra with Online Practice

  • Caryl Lorandini(Author)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  Every counting number greater than 1 is either a prime or a composite number. A prime number has exactly two factors: 1 and the number itself. A composite number has at least three factors. Composite numbers can be broken down into their prime parts or prime factors. Every composite number has a unique set of prime factors. We are going to decompose a composite number into its prime factors.

  Let’s start with 48. Pick any two factors of 48 other than 1 and 48. Keep making branches by doing the same process for the factors you chose. Circle the prime numbers as those are complete. Once all the branches have stopped with circles, you are finished. Gather your numbers for the solution. ANOTHER EXAMPLE: You can always check your answer by multiplying.

  EXAMPLE 8.2

  1)Find the prime factorization of the following: a)36 b)42 c)96 SOLUTIONS

  1)a)22 × 32

  b)2 × 3 × 7

  c)25 × 3

  8.3 What Are Common Factors and Multiples?

  DEFINITIONS

  Common factorA number, polynomial, or quantity that evenly divides into two or more numbers or algebraic expressions.

  Greatest common factor (GCF)The greatest number or expression that is a factor of two or more numbers or expressions.

  MultipleThe product of a given whole number and any other whole number.

  Common multipleA whole number that is a multiple of two or more given numbers.

  Least common multiple (LCM)The smallest number, greater than zero, that is a multiple of two or more numbers.

  Common Factors are really just like they sound: factors that are the same. This is often used in reducing fractions. To find a common factor you can look at both lists of factors. For example: The Common Factors are the factors on both lists: 1, 2, 4. The greatest common factor would be 4, since it is the greatest number on both lists. Another way to find the greatest common factor would be to decompose the numbers to their prime factors. The factors that are common to both lists are 2 × 2, which you multiply to get the GCF of 4. Let’s try it with three numbers: The Common Factors are 1, 2, 3, and 6. The GCF is 6 as it is the greatest number.

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  eBook - ePub

  Math Intervention 3-5

  Building Number Power with Formative Assessments, Differentiation, and Games, Grades 3-5

  • Jennifer Taylor-Cox(Author)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  To use the spinner, place a paper clip in the center. Place pencil point through the paper clip at the center of the spinner. Holding the pencil securely with one hand, spin the paper clip with the other hand.

  Finding Common Factors

  What is the Finding Common Factors Concept?

  A factor of a number will divide that number evenly. For example, 9 is a factor of 18 because 18 can be divided by 9 without any remainders or fractional parts of a number.

  Eighteen has other factors, too. It is also divided evenly by 6 and 3, also by 1 and itself. The factors of 18 are 1, 2, 3, 6, 9, 18. If a number is only divisible by 1 and itself, it is a prime number. Often the task is to find Common Factors of two or more numbers. For example, 5 is a common factor of 30 and 20 because 5 is a factor of 30 and of 20. In fact, 1, 2, 5, 10 are all Common Factors of 30 and 20. If the task is to find the greatest common factor (GCF) of 30 and 20, we compare the factors of 30 and the factors of 20 to find the largest common factor, which is 10.

  CCSS

  Operations and Algebraic Thinking

  Formative Assessment

  To find out if a student understands the finding factors concept ask the student to name at least five factors of the following numbers:

  12

  50

  48

  Then ask the student to find at least two Common Factors of the following pairs of numbers:

  15, 35

  12, 18

  36, 16

  Then ask the student to find the GCF of the following numbers:

  9, 12, 24

  18, 48, 30

  33, 7, 20

  Successful Strategies

  As with finding the LCM, a beneficial way to help students find the GCF is to have students show the factors of each number in a skip counting pattern. When students examine these patterns it is easier to find the Common Factors and the GCF. Students may show the factors with numbers, words, or models. The hundred chart serves as a valuable tool in helping students to find common and greatest Common Factors.

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  eBook - ePub

  Maths from Scratch for Biologists

  • Alan J. Cann(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  5. To factor an integer, break the integer down into a group of numbers whose product equals the original number. Factors are separated by multiplication signs. Note that the number 1 is the factor of every number. All factors of a number can be divided exactly into that number.

  2.6. Lowest common multiple and greatest common factor

  When manipulating fractions you frequently need to find these two terms – you will see examples of this below. To find the lowest common multiple (LCM) of two numbers, make a table of multiples (e.g. 2 and 3; see Table 2.2 ). To find the greatest common factor (GCF), meaning numbers or expressions by which a larger number can be divided exactly (‘factoring’), make a table of factors (e.g. 8 and 12; see Table 2.3 ).

  Table 2.2 Table of multiples

  Multiples of 2	Common multiples	Multiples of 3
  246–	––6–	3–69

  Table 2.3 Table of factors

  Factors of 8	Common Factors	Factors of 12
  12–48–	12–4––	1234612

  2.7. Adding and subtracting fractions

  To be able to add or subtract fractions to or from fractions, the denominators must be the same (‘common’): cannot be added, but can.

  To find a common denominator so you can add or subtract fractions, find the LCM of all the denominators involved. Then, make the denominators equal the LCM by multiplying both the denominator and numerator by the corresponding factor of the LCM. Whenever you manipulate fractions, the final step is to reduce the answer to the lowest terms:

  1. Factor the numerator. 2. Factor the denominator. 3. Find the fraction mix that equals 1.

  Example

  Add

  	The LCM of 3 and 5 (the denominators) is 15.
  	Both denominators must equal the LCM, so multiply 3 by 5, and 5 by 3. Now both denominators are the same (‘common’).
  	To avoid altering the problem, multiply the numerators by the same factor as their respective denominators. This is the same as multiplying each fraction by 15/15, i.e. by 1.
  	Now the denominators the same, add the fractions together.
  	You cannot reduce this fraction further, i.e. reduce numerator and denominator to their LCFs, so this is the final answer.

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  eBook - ePub

  GMAT Advanced Quant

  250+ Practice Problems & Online Resources

  • (Author)
  • 2020(Publication Date)
  • Manhattan Prep

    (Publisher)

  out to the side. These can appear in algebraic or numerical expressions:

  • x18 + 2x16 + x14  → x14 is a factor of each term → x14 (x4 + 2x2 + 1) = x14 (x2 + 1)2

  → get common denominators, then cross them all off →

  Factorials are particularly noteworthy, as they often have an abundance of shared factors. For any integer n, the factorial n! is calculated as follows: n! = n(n − 1)(n − 2)(n − 3) . . . 1. Thus, all the terms in 4! = (4)(3)(2)(1) are also Common Factors of 6! = (6)(5)(4)(3)(2)(1) = (6)(5)(4!).

  More generally, factorials are “super multiples.” Without ever computing their precise value, you can tell that they’re divisible by all sorts of numbers. For example:

  If x is an integer between 7! + 2 and 7! + 4, inclusive, is x prime?

  x is one of the following integers: 7! + 2 = (7)(6)(5)(4)(3)(2)(1) + 2 7! + 3 = (7)(6)(5)(4)(3)(2)(1) + 3 7! + 4 = (7)(6)(5)(4)(3)(2)(1) + 4

  x has one of the following factors, if x is: 7! + 2 = 2 × Integer 7! + 3 = 3 × Integer 7! + 4 = 4 × Integer…so x is not prime!

  Sometimes a common factor is just a random number buried inside a couple of larger numbers. Find it and pull it out:

  10’s in the numerator line up with 5’s in the corresponding digit place of the denominator:

  Try-It #6-3

  If n is a positive integer and is a positive integer, what is the value of n ?

  1. n is prime.
  2. n < 3

  What would need to be true in order for the square root to be a positive integer? The number under the square root symbol would have to be a perfect square. Rearrange the expression to determine whether there are any restrictions that could help narrow down the possibilities before going to the statements. Try to break the numbers down into primes to locate and pull out any existing perfect squares:

  Pull out common terms:

  It turns out that you can also pull out the term 7

  n

  − 1 . The term 7

  n + 1

  = (7

  n − 1

  )(72

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  Effective Techniques to Motivate Mathematics Instruction

  • Alfred Posamentier, Stephen Krulik(Authors)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  Ideal recreations are those that appear difficult and yet are surprisingly simple to resolve. Such solutions can often require thinking “outside the box.” Illustrations can be found in many forms of mathematical thinking and problem processing. In selecting these motivational devices, teachers should be mindful not to make them so easy for the intended population that they become silly, or so difficult that they are beyond the reach of most students. In assessing the appropriateness of these recreational motivational devices, one needs to consider that a student’s emotional development is also influenced by factors that occur within the classroom and can change with the various stimuli provided by the teacher. Thus teachers’ assessments of students should be flexible to anticipate changes throughout the course. As a student develops his or her intellectual capacity, so, too, will the appreciation, understanding, and processing of a puzzle evolve with time.

  Unfortunately too many adults still harbor a fear of or dislike of mathematics largely because their teachers did not take into account the importance of motivating instruction—especially using recreational techniques. This sort of motivational device is an excellent way of not only introducing the lesson, but also demonstrating a light-hearted aspect of mathematics that can have a lasting effect on students.

  Reasoning problems and mathematical games can demonstrate the fun that mathematics can offer. Such activities often win over the uninitiated and serve as much more than mere motivation for the forthcoming lesson.

  Topic: Identifying Factors of Numbers

  Materials or Equipment Needed

  An appropriate medium to display the mathematics.

  Implementation of the Motivation Strategy

  Begin the lesson by asking students to find all the proper factors of 220 (except for 220 itself) and then finding the sum of these factors. They should then do the same with the number 284. If they did this correctly, they should have come up with an unusual result. These two numbers can be considered “friendly numbers”!

  What could possibly make two numbers friendly? Your students’ first reaction might be that these numbers are friendly to them. Remind them that we are talking here about numbers that are “friendly” to each other. Mathematicians have decided that two numbers are considered friendly (or, as often used in the more sophisticated literature, “amicable”) if the sum of the proper divisors of the first number equals the second and the sum of the proper divisors of the second number equals the first.

  Sounds complicated? Have your students now consider the smallest pair of friendly numbers: 220 and 284. They should have gotten the following, which should show the recreational aspect of mathematics.

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