Key Learning Outcomes
By the end of this chapter, “Multiples and Factors”, readers will:
- What are factors and multiples?
- Prime and composite numbers
- Even and odd numbers
- Prime factorization
- Common factors and common multiples
- Highest Common Factor (HCF)
- Divisibility rules
- Practice questions and real-life examples
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Introduction
Have you ever arranged chairs in equal rows or shared candies among friends equally? Then congratulations—you’ve already used multiples and factors in real life!
In Class 5, the chapter “Multiples and Factors” introduces you to how numbers relate to each other through multiplication and division. It is a foundation for many advanced topics like HCF, LCM, fractions, and more.
Let’s understand these concepts in a fun and simple way!
What Are Factors?
A factor of a number divides it exactly without leaving a remainder.
Example:
Factors of 12 are:
1, 2, 3, 4, 6, 12
Why? Because all these numbers divide 12 completely.
- 12 ÷ 1 = 12 (no remainder)
- 12 ÷ 2 = 6 (no remainder)
- 12 ÷ 3 = 4 (no remainder)
- 12 ÷ 4 = 3 (no remainder)
- 12 ÷ 6 = 2 (no remainder)
- 12 ÷ 12 = 1 (no remainder)
Important Points:
- Every number has 1 and itself as factors.
- Factors are always less than or equal to the number.
- A number can have many factors.
Tip: Factors are always less than or equal to the number or To find factors, try dividing the number by 1, 2, 3, 4, etc., until the number itself.
What Are Multiples?
A multiple of a number is what you get when you multiply that number by any whole number.
Example:
Multiples of 4 are:
4, 8, 12, 16, 20, 24, …
(basically 4 × 1, 4 × 2, 4 × 3…)
Because:
- 4 × 1 = 4
- 4 × 2 = 8
- 4 × 3 = 12 and so on.
Important Points:
- Multiples are greater than or equal to the number.
- Every number has infinite multiples.
- A number is always a multiple of itself.
Tip: Multiples are always equal to or more than the number.
Difference Between Factors and Multiples
Feature | Factors | Multiples |
---|---|---|
Meaning | Numbers that divide a number | Numbers that come in the multiplication table |
Quantity | Limited | Unlimited |
Example (6) | 1, 2, 3, 6 | 6, 12, 18, 24, 30, … |
Prime and Composite Numbers
Prime Numbers:
A prime number has only two factors: 1 and itself.
Example:
2, 3, 5, 7, 11, 13, 17, 19, …
Composite Numbers:
A composite number has more than two factors.
Example:
4, 6, 8, 9, 10, 12, 14, 15, …
Note: 1 is neither prime nor composite.
Even and Odd Numbers
Even Numbers:
Even numbers end in 0, 2, 4, 6, 8. They are divisible by 2.
Example:
12, 26, 38, etc.
Odd Numbers:
Odd numbers end in 1, 3, 5, 7, 9. Not divisible by 2.
Example:
15, 37, 49, etc.
Prime Factorization
It is the process of breaking a number into its prime factors.
Example:
Prime factorization of 36:
36 = 2 × 2 × 3 × 3
= 2² × 3²
Common Factors and Common Multiples
Common Factors: Factors shared by two or more numbers
Example:
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 18 = 1, 2, 3, 6, 9, 18
Common = 1, 2, 3, 6
Common Multiples: Multiples shared by two or more numbers
Example:
Multiples of 3 = 3, 6, 9, 12, 15, 18
Multiples of 4 = 4, 8, 12, 16, 20
Common = 12, 24, …
Highest Common Factor (HCF)
What is HCF?
The HCF (Highest Common Factor) of two or more numbers is the largest number that divides each of them exactly.
How to Find HCF?
Method 1: Listing Factors
- List all factors of each number.
- Find the common factors.
- The highest one is the HCF.
Example:
Find HCF of 12 and 18
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common factors: 1, 2, 3, 6
- HCF = 6
Method 2: Prime Factorization
- Write prime factors of both numbers.
- Multiply the common prime factors.
Example:
- 12 = 2 × 2 × 3
- 18 = 2 × 3 × 3
- Common prime factors: 2 × 3 = 6
Method 3: Long Division
LCM – Lowest Common Multiple
What is LCM?
The LCM (Lowest Common Multiple) of two or more numbers is the smallest number that is a multiple of all of them.
How to Find LCM?
Method 1: Listing Multiples
- List first few multiples of each number.
- Find the first common multiple.
Example:
Find LCM of 4 and 5
- Multiples of 4: 4, 8, 12, 16, 20, 24
- Multiples of 5: 5, 10, 15, 20, 25
- LCM = 20
Method 2: Prime Factorization
- Write prime factors of each number.
- Multiply all prime factors, taking the highest power of each.
Example:
- 4 = 2 × 2
- 5 = 5
- LCM = 2 × 2 × 5 = 20
Why Are HCF and LCM Important?
HCF is used when dividing things into smaller sections (like equally sharing sweets).
LCM is used when we want things to happen together (like finding when two bells will ring together).
Divisibility Rules (1 to 10)
Quick rules to check if a number is divisible:
Number | Rule Example |
---|---|
2 | Last digit is even → 826 |
3 | Sum of digits divisible by 3 → 453 (4+5+3=12) |
4 | Last 2 digits divisible by 4 → 832 (32) |
5 | Last digit is 0 or 5 → 495 |
6 | Divisible by both 2 and 3 |
8 | Last 3 digits divisible by 8 |
9 | Sum of digits divisible by 9 |
10 | Last digit is 0 |
Important Facts
- Addition is commutative but subtraction is not.
- Zero is the identity for addition.
- Always align numbers by place value before operations.
- Subtraction result is called difference.
Practice Questions:
A. Find Factors
- Factors of 18 = ______
- Factors of 30 = ______
B. List First 5 Multiples
- Multiples of 6 = ______
- Multiples of 9 = ______
C. Prime or Composite?
- 17 → ______
- 27 → ______
D. Word Problem
- Laiba has 36 pens and 24 pencils. She wants to make equal gift sets using all. What is the maximum number of sets she can make?
Solution: HCF of 36 and 24 = 12 sets
Common Mistakes to Avoid
Confusing factors with multiples
➤ Factors divide a number; multiples are the results of multiplying.
Thinking 1 is prime
➤ It is neither prime nor composite.
Skipping 1 and the number itself in the factor list
Real-Life Applications
- Arranging chairs in rows = using factors
- Timetables of buses/trains = common multiples
- Packing equal gift boxes = HCF
- Creating schedules = LCM (introduced later)
Supplementary Materials:
Provide downloadable materials for learners to review:
- – PDF Guide: “Coming Soon”
- – Cheat Sheet: “Coming Soon”
- – Video Source: “JNG ACADEMY“
- – Articles: “Blog Page“
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