Highest Common Factor

8 Key excerpts on "Highest Common Factor"

• 

  eBook - PDF

  Introductory Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  ∴ 3 is the Highest Common Factor of 12 and 21. Example 2 Find the HCF of 51 and 19. Solution 51 = 1 × 51 51 51 1 19 = 1 × 19 19 19 1 ∴ HCF = 1 • 1 appears as a factor in both 51 and 19. ∴ 1 is the common factor. 79 Introductory Mathematics| Hands-On In some cases the HCF may be 1 or one of the actual numbers. Example 3 Find the HCF of 20 x 3 y 2 ; 15 x 2 y 4 and 25 xy 3 . Solution 20 x 3 y 2 = 2 × 2 × 5 × x × x × x × y × y 15 x 2 y 4 = 5 × 3 × x × x × y × y × y × y 25 xy 3 = 5 × 5 × x × y × y × y ∴ HCF = 5 × x × y × y = 5 xy 2 • 5 xy 2 is the common factor and the biggest expression that can be divided into 20 x 3 y 2 ; 15 x 2 y 4 and 25 xy 3 . The variables with the lowest powers form part of the HCF. Example 4 Find the HCF of 120 x 4 yz 3 ; 45 x 2 y 4 z 2 and 105 x 5 y 3 z 2 . Solution 120 x 4 yz 3 45 x 2 y 4 z 2 105 x 5 y 3 z 2 2 120 3 45 3 105 2 60 315 5 35 2 30 5 5 7 7 3 15 1 1 5 5 1 120 x 4 yz 3 = 2 × 2 × 2 × 3 × 5 × x 4 yz 3 45 x 2 y 4 z 2 = 3 × 3 × 5 × x 2 y 4 z 2 105 x 5 y 3 z 2 = 3 × 5 × 7 × x 5 y 3 z 2 ∴ HCF = 3 × 5 × x × x × y × z × z = 15 x 2 yz 2 • 15 x 2 yz 2 is the common factor and the biggest expression that can be divided into 120 x 4 yz 3 ; 45 x 2 y 4 z 2 and 105 x 5 y 3 z 2 • The variable with the lowest power, x 2 yz 2 , is part of the HCF. 80 Module 3 • Factorisation and fractions Activity 3.1 1. Find the prime factors of the following. a) 16 b) 98 c) 175 d) 144 xy 2 e) 360 a 3 b 2 2. Determine the HCF of the following. a) 12; 14 and 16 b) 128; 24 and 40 c) 32 x 3 y 3 ; 48 x 2 y and 70 xy d) 28 a 3 b 3 ; 49 ab 3 and 14 a 2 b 4 e) 64 x 2 y 4 z 4 ; 81 x 2 y 3 z 2 and 25 xy 2 z 5 f) 108 mn 2 ; 80 m 2 n 2 and 72 mn g) (2 xyz ) 2 ; 4 xy 2 z 2 and 8 xz 3 81 Introductory Mathematics| Hands-On 3.1.3 Lowest common multiple The lowest common multiple ( LCM ) of two or more numbers is the smallest possible number into which all the numbers will divide exactly, for example the LCM of 3, 4 and 8 is 24.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  N1 Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  ∴ 3 is the Highest Common Factor of 12 and 21. Example 2 Find the HCF of 51 and 19. Solution 51 = 1 × 51 51 51 1 19 = 1 × 19 19 19 1 ∴ HCF = 1 • 1 appears as a factor in both 51 and 19. ∴ 1 is the common factor. 79 N1 Mathematics| Hands-On In some cases the HCF may be 1 or one of the actual numbers. Example 3 Find the HCF of 20 x 3 y 2 ; 15 x 2 y 4 and 25 xy 3 . Solution 20 x 3 y 2 = 2 × 2 × 5 × x × x × x × y × y 15 x 2 y 4 = 5 × 3 × x × x × y × y × y × y 25 xy 3 = 5 × 5 × x × y × y × y ∴ HCF = 5 × x × y × y = 5 xy 2 • 5 xy 2 is the common factor and the biggest expression that can be divided into 20 x 3 y 2 ; 15 x 2 y 4 and 25 xy 3 . The variables with the lowest powers form part of the HCF. Example 4 Find the HCF of 120 x 4 yz 3 ; 45 x 2 y 4 z 2 and 105 x 5 y 3 z 2 . Solution 120 x 4 yz 3 45 x 2 y 4 z 2 105 x 5 y 3 z 2 2 120 3 45 3 105 2 60 315 5 35 2 30 5 5 7 7 3 15 1 1 5 5 1 120 x 4 yz 3 = 2 × 2 × 2 × 3 × 5 × x 4 yz 3 45 x 2 y 4 z 2 = 3 × 3 × 5 × x 2 y 4 z 2 105 x 5 y 3 z 2 = 3 × 5 × 7 × x 5 y 3 z 2 ∴ HCF = 3 × 5 × x × x × y × z × z = 15 x 2 yz 2 • 15 x 2 yz 2 is the common factor and the biggest expression that can be divided into 120 x 4 yz 3 ; 45 x 2 y 4 z 2 and 105 x 5 y 3 z 2 • The variable with the lowest power, x 2 yz 2 , is part of the HCF. 80 Module 3 • Factorisation and fractions Activity 3.1 1. Find the prime factors of the following. a) 16 b) 98 c) 175 d) 144 xy 2 e) 360 a 3 b 2 2. Determine the HCF of the following. a) 12; 14 and 16 b) 128; 24 and 40 c) 32 x 3 y 3 ; 48 x 2 y and 70 xy d) 28 a 3 b 3 ; 49 ab 3 and 14 a 2 b 4 e) 64 x 2 y 4 z 4 ; 81 x 2 y 3 z 2 and 25 xy 2 z 5 f) 108 mn 2 ; 80 m 2 n 2 and 72 mn g) (2 xyz ) 2 ; 4 xy 2 z 2 and 8 xz 3 81 N1 Mathematics| Hands-On 3.1.3 Lowest common multiple The lowest common multiple ( LCM ) of two or more numbers is the smallest possible number into which all the numbers will divide exactly, for example the LCM of 3, 4 and 8 is 24.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Mathematical Practices, Mathematics for Teachers

  Activities, Models, and Real-Life Examples

  • Ron Larson, Robyn Silbey(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The notation for the greatest common factor of a and b is written as GCF(a, b). To find the GCF(a, b), write the prime factorizations of a and b. The greatest common factor is the product of the common factors. Example a = 20, b = 12 a = 20 b = 12 2 2 5 3 Common factors a = 2 ⋅ 2 ⋅ 5 b = 2 ⋅ 2 ⋅ 3 GCF(20, 12) = 2 ⋅ 2 = 4 If a divides b, then GCF(a, b) = a. If b divides a, then GCF(a, b) = b. Example GCF(3, 12) = 3 EXAMPLE 1 Finding the Greatest Common Factor Find the greatest common factor of 16 and 24. SOLUTION Begin by writing the prime factorization of each number. a = 16: 16 b = 24: 24 4 4 4 6 2 2 2 2 2 2 2 3 So, 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2 and 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3. Next, use a diagram to organize the prime factors of a and b. a = 16 b = 24 2 2 2 2 3 Common factors The greatest common factor of 16 and 24 is 2 ⋅ 2 ⋅ 2 = 8. When you are teaching strategies for finding the GCF of two whole numbers, remind your students to first think about the numbers. 1. If both are primes, then the GCF is 1. GCF(3, 5) = 1 2. If one number is a multiple of the other, then the GCF is the lesser of the two numbers. GCF(3, 15) = 3 3. If one of the numbers is 0, then the GCF is the greater of the two numbers because 0 is divisible by any nonzero number. GCF(3, 0) = 3 Classroom Tip Standards Grades 3–5 Operations and Algebraic Thinking Students should gain familiarity with factors and multiples. Grades 6–8 The Number System Students should find common factors and multiples. 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.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  The Higher Arithmetic

  An Introduction to the Theory of Numbers

  • H. Davenport(Author)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  Two numbers are said to be relatively prime ∗ if they have no common divisor except 1, or in other words if their H.C.F. is 1. This will be the case if and only if the last remainder, when Euclid’s algorithm is applied to the two numbers, is 1. ∗ This is, of course, the same definition as in §5, but is repeated here because the present treatment is independent of that given previously. 18 The Higher Arithmetic 7. Another proof of the fundamental theorem We shall now use Euclid’s algorithm to give another proof of the funda- mental theorem of arithmetic, independent of that given in §4. We begin with a very simple remark, which may be thought to be too obvious to be worth making. Let a, b, n be any natural numbers. The high- est common factor of na and nb is n times the Highest Common Factor of a and b. However obvious this may seem, the reader will find that it is not easy to give a proof of it without using either Euclid’s algorithm or the fundamental theorem of arithmetic. In fact the result follows at once from Euclid’s algorithm. We can sup- pose a > b. If we divide na by nb, the quotient is the same as before (namely q ) and the remainder is nc instead of c. The equation (2) is replaced by na = q .nb + nc. The same applies to the later equations; they are all simply multiplied throughout by n. Finally, the last remainder, giving the H.C.F. of na and nb, is nh, where h is the H.C.F. of a and b. We apply this simple fact to prove the following theorem, often called Euclid’s theorem, since it occurs as Prop. 30 of Book VII. If a prime divides the product of two numbers, it must divide one of the numbers (or possibly both of them). Suppose the prime p divides the product na of two numbers, and does not divide a. The only factors of p are 1 and p, and therefore the only common factor of p and a is 1. Hence, by the theorem just proved, the H.C.F. of np and na is n. Now p divides np obviously, and divides na by hypothesis.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introductory Algebra

  Concepts with Applications

  • Charles P. McKeague(Author)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Note As you will see as we progress through the book, factoring is a tool that is used in solving a number of problems. Before seeing how it is used, however, we first must learn how to do it. So, in this section and the sections that follow, we will be developing our factoring skills. DEFINITION greatest common factor The greatest common factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. Image © sxc.hu, Kaliyoda, 2009 KEY WORDS 385 5.1 The Greatest Common Factor and Factoring by Grouping EXAMPLE 1 Find the greatest common factor for the polynomial 3x 5 + 12x 2 . Solution The terms of the polynomial are 3x 5 and 12x 2 . The largest number that divides the coefficients is 3, and the highest power of x that is a factor of x 5 and x 2 is x 2 . Therefore, the greatest common factor for 3x 5 + 12x 2 is 3x 2 ; that is, 3x 2 is the largest monomial that divides each term of 3x 5 + 12x 2 . EXAMPLE 2 Find the greatest common factor for 8a 3 b 2 + 16a 2 b 3 + 20a 3 b 3 . Solution The largest number that divides each of the coefficients is 4. The highest power of the variable that is a factor of a 3 b 2 , a 2 b 3 , and a 3 b 3 is a 2 b 2 . The greatest common factor for 8a 3 b 2 + 16a 2 b 3 + 20a 3 b 3 is 4a 2 b 2 . It is the largest monomial that is a factor of each term. Once we have recognized the greatest common factor of a polynomial, we can apply the distributive property and factor it out of each term. We rewrite the polynomial as the product of its greatest common factor with the polynomial that remains after the greatest common factor has been factored from each term in the original polynomial. EXAMPLE 3 Factor the greatest common factor from 3x − 15. Solution The greatest common factor for the terms 3x and 15 is 3. We can rewrite both 3x and 15 so that the greatest common factor 3 is showing in each term. It is important to realize that 3x means 3 ⋅ x. The 3 and the x are not “stuck” together.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Elementary Algebra 2e

  • Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
  • 2020(Publication Date)
  • Openstax

    (Publisher)

  Find the Greatest Common Factor of Two or More Expressions Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring. We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM. Greatest Common Factor The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions. First we’ll find the GCF of two numbers. EXAMPLE 7.1 HOW TO FIND THE GREATEST COMMON FACTOR OF TWO OR MORE EXPRESSIONS Find the GCF of 54 and 36. Solution 808 Chapter 7 Factoring This OpenStax book is available for free at http://cnx.org/content/col31130/1.4 Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18. 54 = 18 · 3 36 = 18 · 2 TRY IT : : 7.1 Find the GCF of 48 and 80. TRY IT : : 7.2 Find the GCF of 18 and 40. We summarize the steps we use to find the GCF below. In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor. EXAMPLE 7.2 Find the greatest common factor of 27x 3 and 18x 4 . Solution Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors. The GCF of 27x 3 and 18x 4 is 9x 3 . TRY IT : : 7.3 Find the GCF: 12x 2 , 18x 3 . TRY IT : : 7.4 Find the GCF: 16y 2 , 24y 3 . EXAMPLE 7.3 Find the GCF of 4x 2 y, 6xy 3 . HOW TO : : FIND THE GREATEST COMMON FACTOR (GCF) OF TWO EXPRESSIONS. Factor each coefficient into primes. Write all variables with exponents in expanded form. List all factors—matching common factors in a column.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  C++ for Mathematicians

  An Introduction for Students and Professionals

  • Edward Scheinerman(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 3 Greatest Common Divisor 3.1 The problem For integers a and b , the greatest common divisor of a and b is the largest integer d such that d is a factor of both a and b . The greatest common divisor is denoted gcd ( a , b ) . We say a and b are relatively prime provided gcd ( a , b ) = 1. In this chapter we develop many C++ concepts by studying a particular problem involving the gcd of integers. Here is the classic problem: Let n be a positive integer. Two integers, a and b , are selected independently and uniformly from the set { 1 , 2 ,..., n } . Let p n be the probability that a and b are relatively prime. Does lim n → ∞ p n exist and, if so, what is its value? The computer cannot solve this problem for us, but it can help us to formulate a conjecture. We try a number of approaches including exhaustive enumeration, generating pairs of numbers at random and recording the results, and the use of Euler’s totient. The first order of business, however, is to develop a procedure to compute the greatest common divisor of two integers. 3.2 A first approach In this section we develop a correct, but highly inefficient, procedure for calculat-ing the greatest common divisor of two integers. Our goal is to introduce a number of C++ concepts as well as to create the gcd procedure. This procedure takes two integers and returns their greatest common divisor. Later, we replace this inefficient procedure with a much more efficient method. Before we begin, however, we need to address a bit of terminology. Mathemati-cians and computer programmers use the word function differently. A mathemati-cian’s function assigns to each value x a unique value y = f ( x ) . Suppose we calculate, say f ( 8 ) and the result is 17. Then if we calculate f ( 8 ) a few minutes later, we are guaranteed that the result is still 17. 31

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Basic Mathematics for College Students with Early Integers

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Now Try Problem 49 Copyright 2019 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. CHAPTER 1 • Whole Numbers 92 OBJECTIVE 4 Find the GCF using prime factorization. We can find the GCF of two (or more) numbers by listing the factors of each number. However, this method can be lengthy. Another way to find the GCF uses the prime factorization of each number. In Example 7, we found that the GCF of 18 and 45 is 9. Note that 9 is the greatest number that divides 18 and 45. 18 9  2 45 9  5 In general, the greatest common factor of two (or more) numbers is the largest number that divides them exactly. For this reason, the greatest common factor is also known as the greatest common divisor (GCD) and we can write GCD (18, 45)  9. Finding the GCF Using Prime Factorization To find the greatest common factor of two (or more) whole numbers: 1. Prime factor each number. 2. Identify the common prime factors. 3. The GCF is a product of all the common prime factors found in Step 2. If there are no common prime factors, the GCF is 1. Find the GCF of 48 and 72. Strategy We will begin by finding the prime factorizations of 48 and 72. WHY Then we can identify any prime factors that they have in common. Solution Step 1: Prime factor 48 and 72. 48  2 ? 2 ? 2 ? 2 ? 3 72  2 ? 2 ? 2 ? 3 ? 3 Step 2: The circling shows that 48 and 72 have four common prime factors: Three common factors of 2 and one common factor of 3. Step 3: The GCF is the product of the circled prime factors. GCF (48, 72)  2 ? 2 ? 2 ? 3  24 EXAMPLE 8 Find the GCF of 36 and 60.

  Sign up to read

  Learn more about book