Factors, Multiples, LCM and HCF

Week 3: Factors, Multiples, LCM and HCF

Class: JSS1 | Subject: Mathematics | Term: First Term

Learning Objectives

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

• Define and identify factors and multiples of whole numbers
• Distinguish between prime and composite numbers
• Find prime factors using factor trees
• Calculate the Highest Common Factor (HCF) of two or more numbers
• Calculate the Lowest Common Multiple (LCM) of two or more numbers

What are Factors?

Factors are numbers that divide exactly into another number without leaving a remainder. Every number has at least two factors: 1 and itself.

Example: Find the factors of 24

24 / 1 = 24, 24 / 2 = 12, 24 / 3 = 8, 24 / 4 = 6
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Example: Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

What are Multiples?

Multiples are the products obtained when a number is multiplied by the counting numbers (1, 2, 3, 4, ...). Every number has infinitely many multiples.

Example: First 10 multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

Prime and Composite Numbers

A prime number is a number greater than 1 that has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37. Note that 2 is the only even prime number.

A composite number is a number greater than 1 that has more than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15. The number 1 is neither prime nor composite.

Prime Factorization

Prime factorization is the process of expressing a composite number as a product of its prime factors. Every composite number can be expressed uniquely as a product of prime numbers (Fundamental Theorem of Arithmetic).

Example: Find the prime factorization of 60

60 / 2 = 30, 30 / 2 = 15, 15 / 3 = 5, 5 / 5 = 1
Therefore, 60 = 2 x 2 x 3 x 5 = 2 squared x 3 x 5

Example: Prime factorization of 84

84 / 2 = 42, 42 / 2 = 21, 21 / 3 = 7, 7 / 7 = 1
Therefore, 84 = 2 squared x 3 x 7

Highest Common Factor (HCF)

The HCF (also called Greatest Common Divisor) of two or more numbers is the largest factor that divides all the numbers exactly.

Method 1: Listing Factors

Find the HCF of 24 and 36:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12
HCF = 12

Method 2: Prime Factorization

Find the HCF of 48 and 72:
48 = 2x2x2x2 x 3
72 = 2x2x2 x 3x3
HCF = Take lowest power of each common prime = 2x2x2 x 3 = 24

Lowest Common Multiple (LCM)

The LCM of two or more numbers is the smallest number that is a multiple of all the given numbers.

Method 1: Listing Multiples

Find the LCM of 6 and 8:
Multiples of 6: 6, 12, 18, 24, 30...
Multiples of 8: 8, 16, 24, 32...
LCM = 24

Method 2: Prime Factorization

Find the LCM of 12 and 18:
12 = 2x2 x 3
18 = 2 x 3x3
LCM = Take highest power of each prime = 2x2 x 3x3 = 36

Example: Find the LCM and HCF of 15, 20, and 30

15 = 3 x 5, 20 = 2x2 x 5, 30 = 2 x 3 x 5
HCF = 5 (only common factor)
LCM = 2x2 x 3 x 5 = 60

Relationship Between LCM and HCF

For any two numbers a and b: LCM(a,b) x HCF(a,b) = a x b

Example: Verify for 12 and 18: LCM=36, HCF=6. 36 x 6 = 216. 12 x 18 = 216. Correct!

Word Problems

Problem 1: Two bells ring at intervals of 12 minutes and 18 minutes respectively. If they ring together at 9:00 AM, when will they next ring together?
Solution: LCM of 12 and 18 = 36 minutes. They will ring together at 9:36 AM.

Problem 2: A teacher wants to divide 48 pencils and 36 erasers equally among students with no items left over. What is the maximum number of students?
Solution: HCF of 48 and 36 = 12. Maximum 12 students (each gets 4 pencils and 3 erasers).

Evaluation Questions

• 1. List all the factors of 72
• 2. Find the prime factorization of 180
• 3. Find the HCF of 42 and 56
• 4. Find the LCM of 15 and 25
• 5. Three traffic lights flash at intervals of 8, 12, and 15 seconds. If they flash together, after how many seconds will they flash together again?

Summary

Factors divide into a number exactly, while multiples result from multiplying a number by counting numbers. Prime numbers have exactly two factors, and composite numbers have more than two. The HCF is the largest common factor of two or more numbers, while the LCM is the smallest common multiple. Both can be found efficiently using prime factorization. These concepts are essential for working with fractions, ratios, and many real-life problems.