HomeJSS 1MathematicsLowest Common Multiple (L.C.M) Highest Common Factor (H.C.F) of Whole Numbers Mathematics JSS 1 First Term Lesson Notes Week 3

Lowest Common Multiple (L.C.M) Highest Common Factor (H.C.F) of Whole Numbers Mathematics JSS 1 First Term Lesson Notes Week 3

Mathematics JSS 1 First Term Lesson Notes Week 3

Subject: Mathematics
Class: JSS 1
Term: First Term
Week: 3
Age: 11-12 years
Topic: Lowest Common Multiple (L.C.M.) and Highest Common Factor (H.C.F.) of Whole Numbers
Sub-topic:

  1. Concepts of L.C.M. and H.C.F.
  2. L.C.M. and H.C.F. by Inspection and by Formula
  3. Solving Problems using L.C.M. and H.C.F.

Duration: 40 minutes

Behavioural Objectives:

By the end of the lesson, students should be able to:

  1. Understand the concepts of L.C.M. and H.C.F.
  2. Find the L.C.M. and H.C.F. of numbers by inspection and formula.
  3. Apply L.C.M. and H.C.F. to solve problems.

Keywords:

  • L.C.M.
  • H.C.F.
  • Multiple
  • Factor
  • Formula

Set Induction (5 minutes):

The teacher introduces the topic by explaining situations where we need to find the least common multiple and greatest common factor in everyday life, such as planning events at regular intervals or sharing items equally among people.

Entry Behaviour:

Students have basic knowledge of multiplication and division of numbers.

Learning Resources and Materials:

  1. Multiplication charts
  2. Worksheets with exercises on L.C.M. and H.C.F.
  3. Flashcards showing prime numbers

Building Background/Connection to Prior Knowledge:

Students have previously learned about multiples and factors of numbers. The concept of L.C.M. and H.C.F. builds on their understanding of these terms.

Embedded Core Skills:

  • Problem-solving
  • Numeracy
  • Logical reasoning
  • Analytical thinking

Instructional Materials:

  1. Chalkboard or whiteboard
  2. Multiplication charts
  3. Flashcards
  4. Worksheets with examples of L.C.M. and H.C.F. problems

Content:

1. Concepts of L.C.M. and H.C.F.:

  • L.C.M. (Lowest Common Multiple):
    The smallest number that is a multiple of two or more numbers.

    • Example: The L.C.M. of 4 and 6 is 12 because 12 is the smallest number that both 4 and 6 can divide without leaving a remainder.
  • H.C.F. (Highest Common Factor):
    The largest number that divides two or more numbers exactly without leaving a remainder.

    • Example: The H.C.F. of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18.

2. L.C.M. and H.C.F. by Inspection and Formulae:

  • Finding L.C.M. by Inspection (Listing multiples):
    • Example: Find the L.C.M. of 8 and 12 by listing the multiples.
      Multiples of 8: 8, 16, 24, 32, 40…
      Multiples of 12: 12, 24, 36, 48…
      The L.C.M. of 8 and 12 is 24 (smallest common multiple).
  • Finding H.C.F. by Inspection (Listing factors):
    • Example: Find the H.C.F. of 16 and 24 by listing the factors.
      Factors of 16: 1, 2, 4, 8, 16
      Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
      The H.C.F. of 16 and 24 is 8 (largest common factor).
  • Finding L.C.M. and H.C.F. by Prime Factorization (Using Formula):
    • L.C.M. Formula:
      L.C.M. = Product of prime factors raised to their highest powers.
      Example: Find the L.C.M. of 18 and 24.
      Prime factorization of 18 = 2 × 3²
      Prime factorization of 24 = 2³ × 3
      L.C.M. = 2³ × 3² = 72
    • H.C.F. Formula:
      H.C.F. = Product of common prime factors raised to their lowest powers.
      Example: Find the H.C.F. of 18 and 24.
      Prime factorization of 18 = 2 × 3²
      Prime factorization of 24 = 2³ × 3
      H.C.F. = 2 × 3 = 6

3. Solving Problems Using L.C.M. and H.C.F.:

  • Problem 1 (L.C.M. Application):
    Two bells ring every 4 minutes and 6 minutes, respectively. At what time will both bells ring together again?
    Solution: Find the L.C.M. of 4 and 6, which is 12. Therefore, the bells will ring together every 12 minutes.
  • Problem 2 (H.C.F. Application):
    A teacher wants to divide 24 pencils and 36 pens equally among students. What is the greatest number of students the teacher can divide the items among?
    Solution: Find the H.C.F. of 24 and 36, which is 12. Therefore, the teacher can divide the pencils and pens among 12 students.

Least Common Multiple and Highest Common Factor of Whole Numbers

Content

  • Multiples
  • Common Multiples
  • Least Common Multiples (LCM)
  • Highest Common Factors (HCF)

Multiples

Definition: A multiple is what you get when you multiply a number by any whole number. Every number has an infinite number of multiples.

Examples:

  • The first four multiples of 12 are:
    • 12 × 1 = 12
    • 12 × 2 = 24
    • 12 × 3 = 36
    • 12 × 4 = 48

Next Five Multiples:

  1. 4: 8, 12, 16, 20, 24
  2. 8: 16, 24, 32, 40, 48
  3. 11: 22, 33, 44, 55, 66

Evaluation:

  1. Find the next 7 multiples of:
    • (a) 15
    • (b) 25
    • (c) 13
  2. Determine which of the following are:
    • (a) multiples of 2
    • (b) multiples of 3
    • (c) multiples of 5
    • (d) multiples of 17

Common Multiples

Definition: Common multiples are multiples shared by two or more numbers.

Examples:

  • Find common multiples of 4, 6, and 8:
    • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48
    • Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48
    • Multiples of 8: 8, 16, 24, 32, 40, 48

    The first two common multiples are 24 and 48.

  • Find common multiples of 5 and 6:
    • Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60
    • Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

    The first three common multiples are 30, 60, and 90.

Evaluation: Find the first four common multiples of:

  1. (a) 4 and 7
  2. (b) 2, 5, and 7
  3. (c) 3, 6, and 9

Least Common Multiple (LCM)

Definition: The LCM of two or more numbers is the smallest multiple that is common to all the numbers.

Finding LCM:

  • Method 1: List multiples until you find a common multiple.
  • Method 2: Use prime factorization.

Examples:

  1. LCM of 24 and 15:
    • Multiples of 24: 24, 48, 72, 96, 120
    • Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120

    The LCM is 120.

  2. Prime Factors Method:
    • LCM of 24 and 15:
      • 24 = 2 × 2 × 2 × 3
      • 15 = 3 × 5
      • LCM = 2 × 2 × 2 × 3 × 5 = 120
    • LCM of 8 and 45:
      • 8 = 2 × 2 × 2
      • 45 = 3 × 3 × 5
      • LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360

Evaluation:

  1. Find the LCM of:
    • (a) 4, 6, and 9
    • (b) 6, 8, 10, and 12
    • (c) 9, 10, 12, and 15
    • (d) 108 and 360
  2. Find the LCM of the following using index notation:
    • (a) 2^3 × 3^2
    • (b) 2^2 × 3 × 5
    • (c) 2^2 × 3^2 × 5

Highest Common Factor (HCF)

Definition: The HCF is the largest number that divides exactly into all the given numbers.

Examples:

  1. HCF of 21 and 84:
    • 21 = 3 × 7
    • 84 = 2 × 2 × 3 × 7
    • HCF = 3 × 7 = 21
  2. HCF of 195 and 330:
    • 195 = 3 × 5 × 13
    • 330 = 2 × 3 × 5 × 11
    • HCF = 3 × 5 = 15
  3. HCF of 288, 180, and 108:
    • 288 = 2 × 2 × 2 × 2 × 2 × 3 × 3
    • 180 = 2 × 2 × 3 × 3 × 5
    • 108 = 2 × 2 × 3 × 3 × 3
    • HCF = 2 × 2 × 3 × 3 = 36

Evaluation:

  1. Find the HCF of:
    • (a) 160, 96, and 224
    • (b) 189, 279, and 108
    • (c) 126, 234, and 90
  2. Find the HCF using index notation:
    • (a) 2^2 × 3^3 × 5
    • (b) 2^2 × 3^3 × 7
    • (c) 2^5 × 3^2 × 5^2 × 7

Weekend Assignment

  1. The value of 2^3 × 3^2 is:
    • (a) 1296
    • (b) 658
    • (c) 729
    • (d) 432
    • (e) 54
  2. The LCM of 12 and 15 is:
    • (a) 90
    • (b) 60
    • (c) 30
    • (d) 120
    • (e) 180
  3. The HCF of 63 and 90 is:
    • (a) 7
    • (b) 3
    • (c) 12
    • (d) 6
    • (e) 9
  4. The first three common multiples of 3 and 11 are:
    • (a) 3, 33, 66
    • (b) 11, 33, 66
    • (c) 33, 66, 99
    • (d) 33, 44, 55
    • (e) 33, 22, 11
  5. Which of the following are multiples of 5?
    • (a) 11, 22, 35
    • (b) 35, 40, 75, 105
    • (c) 54, 35, 40, 75, 105
    • (d) 35, 54, 40, 75
    • (e) 105, 75, 40, 35, 54

Theory

  1. Find the first five multiples of:
    • (a) 5
    • (b) 7
    • (c) 11

    Write down four common multiples of:

    • (a) 3, 4, and 5
    • (b) 3, 10, and 15
  2. Find the LCM of:
    • (a) 9, 24, 32, and 90
    • (b) 2^3 × 5 × 7
    • (c) 3^3 × 7^2
    • (d) 2^4 × 3 × 7^2
  3. Find the HCF of:
    • (a) 160, 96, and 224
    • (b) 189, 279, and 108
    • (c) 126, 234, and 90

    Find the HCF using index notation:

    • (a) 2^2 × 3^3 × 5
    • (b) 2^2 × 3^3 × 7
    • (c) 2^5 × 3^2 × 5^2 × 7

15 Fill-in-the-Blank Questions with Options:

  1. The L.C.M. of 4 and 6 is ______.
    a) 12
    b) 18
    c) 24
    d) 6
  2. The H.C.F. of 18 and 24 is ______.
    a) 6
    b) 8
    c) 4
    d) 12
  3. The L.C.M. of 3 and 9 is ______.
    a) 6
    b) 9
    c) 18
    d) 12
  4. The largest number that divides both 15 and 25 without leaving a remainder is called the ______.
    a) L.C.M.
    b) Prime number
    c) H.C.F.
    d) Multiple
  5. The smallest multiple common to 5 and 10 is the ______.
    a) L.C.M.
    b) H.C.F.
    c) Prime factor
    d) Product
  6. The prime factorization of 24 is ______.
    a) 2³ × 3
    b) 2² × 3²
    c) 2 × 3
    d) 2³ × 5
  7. The H.C.F. of 16 and 20 is ______.
    a) 2
    b) 4
    c) 8
    d) 10
  8. The L.C.M. of 12 and 15 is ______.
    a) 30
    b) 60
    c) 45
    d) 75
  9. The H.C.F. of 30 and 45 is ______.
    a) 5
    b) 10
    c) 15
    d) 20
  10. The L.C.M. of 7 and 14 is ______.
    a) 21
    b) 28
    c) 7
    d) 14
  11. The prime factors of 18 are ______.
    a) 2 × 3²
    b) 3² × 5
    c) 2² × 3
    d) 2 × 5
  12. The L.C.M. of 6 and 9 is ______.
    a) 12
    b) 18
    c) 24
    d) 6
  13. The H.C.F. of 48 and 60 is ______.
    a) 24
    b) 16
    c) 12
    d) 8
  14. The lowest multiple of 5 and 20 is ______.
    a) 40
    b) 60
    c) 20
    d) 10
  15. The L.C.M. of 8 and 16 is ______.
    a) 8
    b) 16
    c) 32
    d) 64

15 FAQs with Answers:

  1. What is L.C.M.?
    The Lowest Common Multiple is the smallest number that two or more numbers can divide exactly.
  2. What is H.C.F.?
    The Highest Common Factor is the largest number that divides two or more numbers without a remainder.
  3. How do we find the L.C.M. of two numbers?
    By listing the multiples of each number and finding the smallest common one.
  4. How do we find the H.C.F. of two numbers?
    By listing the factors of each number and finding the largest common one.
  5. Can the L.C.M. be smaller than the H.C.F.?
    No, the L.C.M. is always greater than or equal to the H.C.F.
  6. Why is L.C.M. useful?
    L.C.M. is useful when we need to find a common time or cycle for two repeating events.
  7. Why is H.C.F. important?
    H.C.F. is important when we need to divide things equally or share items.
  8. What is prime factorization?
    Prime factorization is expressing a number as a product of its prime factors.
  9. How do you find the L.C.M. by prime factorization?
    Multiply all prime factors, using the highest power of each.
  10. How do you find the H.C.F. by prime factorization?
    Multiply the common prime factors, using their lowest power.
  11. What is the L.C.M. of 5 and 10?
    The L.C.M. of 5 and 10 is 10.
  12. What is the H.C.F. of 20 and 30?
    The H.C.F. of 20 and 30 is 10.
  13. How can we use the L.C.M. to solve real-life problems?
    It helps to find when two or more events will happen at the same time.
  14. What is the L.C.M. of 9 and 12?
    The L.C.M. of 9 and 12 is 36.
  15. What is the H.C.F. of 24 and 32?
    The H.C.F. of 24 and 32 is 8.

Presentation:

Step 1: The teacher revises the previous lesson on factors and multiples.
Step 2: The teacher explains the concepts of L.C.M. and H.C.F., using examples.
Step 3: The teacher allows the students to solve L.C.M. and H.C.F. problems in groups and individually.

Teacher’s Activities:

  • Explain the steps for finding L.C.M. and H.C.F. by inspection and by using prime factorization.
  • Provide examples of real-life situations where L.C.M. and H.C.F. are used.
  • Supervise students while they solve L.C.M. and H.C.F. problems.

Learners’ Activities:

  • Participate in discussions about L.C.M. and H.C.F.
  • Work in pairs to find the L.C.M. and H.C.F. of given numbers.
  • Solve the given classwork on L.C.M. and H.C.F.

Assessment:

Solve the following problems:

  1. Find the L.C.M. of 5 and 7.
  2. Find the H.C.F. of 36 and 60.
  3. The L.C.M. of 4 and 10 is ______.
  4. The H.C.F. of 15 and 45 is ______.
  5. The prime factorization of 30 is ______.

Evaluation Questions:

  1. What is the L.C.M. of 3 and 5?
  2. How do you find the H.C.F. of two numbers?
  3. What is the H.C.F. of 40 and 60?
  4. Explain how to find the L.C.M. by prime factorization.
  5. What is the L.C.M. of 6 and 9?

Conclusion:

The teacher goes around to mark students’ work, provides feedback, and corrects any mistakes.

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