Understand what factors and multiples of a number are
Find all factors of a given number
List multiples of a given number
Identify prime numbers and composite numbers
Find the prime factorisation of a number using factor tree and division method
Find common factors and the Highest Common Factor (HCF) of two or more numbers
Find common multiples and the Lowest Common Multiple (LCM) of two or more numbers
Understand the relationship between HCF and LCM
Solve word problems involving HCF and LCM in real-life Indian contexts
Introduction — Where Do We Need HCF and LCM?
Imagine Priya has two ribbons — one is 24 cm long and the other is 36 cm long. She wants to cut both ribbons into pieces of equal length with no ribbon wasted. What is the longest piece she can cut? This is a problem where we need the Highest Common Factor (HCF)!
Now think about this: Arjun goes to the library every 4 days, and Meena goes every 6 days. They both went today. After how many days will they visit the library on the same day again? This is a problem where we need the Lowest Common Multiple (LCM)!
In Class 4, you learned about factors and multiples. In Class 5, we will go deeper — learning about prime factorisation, HCF, and LCM. These are powerful tools used in everyday life: dividing things equally, scheduling events, tiling floors, and much more.
What Are Factors?
A factor of a number is a whole number that divides it exactly (without leaving a remainder).
When we say "6 is a factor of 24," it means 24 ÷ 6 = 4 (exact division, remainder = 0).
Properties of Factors
1 is a factor of every number.
Every number is a factor of itself.
The smallest factor of any number is always 1.
The largest factor of any number is the number itself.
Every factor of a number is less than or equal to that number.
The number of factors of any number is finite (limited).
How to Find All Factors of a Number
To find all factors of a number, we check which numbers from 1 to that number divide it exactly. A quick way is to find factor pairs — pairs of numbers that multiply to give the number.
All factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 (10 factors)
What Are Multiples?
A multiple of a number is the result of multiplying that number by any whole number (1, 2, 3, 4, ...).
For example, multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, ... (5×1, 5×2, 5×3, ...)
Properties of Multiples
Every number is a multiple of 1.
Every number is a multiple of itself (it is the first multiple).
The smallest multiple of any number is the number itself.
Multiples of a number are always greater than or equal to the number.
The number of multiples of any number is infinite (unlimited) — they go on forever!
0 is a multiple of every number (but we usually start listing from the number itself).
Example 1: First 10 multiples of 6:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60
Example 2: First 8 multiples of 9:
9, 18, 27, 36, 45, 54, 63, 72
Example 3: First 8 multiples of 12:
12, 24, 36, 48, 60, 72, 84, 96
Difference Between Factors and Multiples
Property
Factors
Multiples
Definition
Numbers that divide a number exactly
Numbers obtained by multiplying a number
Count
Finite (limited)
Infinite (unlimited)
Smallest
Always 1
The number itself
Largest
The number itself
No largest (goes on forever)
Relation to number
Less than or equal to the number
Greater than or equal to the number
Example (for 12)
1, 2, 3, 4, 6, 12
12, 24, 36, 48, 60, ...
Think About It: If 4 is a factor of 20, then 20 is a multiple of 4. Factors and multiples are like two sides of the same coin! When we say "A is a factor of B," we can also say "B is a multiple of A."
Prime Numbers and Composite Numbers
Numbers can be classified based on how many factors they have:
Definitions
Prime Number: A number that has exactly 2 factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13...
Composite Number: A number that has more than 2 factors. Examples: 4, 6, 8, 9, 10, 12...
Special case — 1: The number 1 has only 1 factor (itself). It is neither prime nor composite.
Smallest prime number: 2 (it is also the only even prime number!)
The Sieve of Eratosthenes is a method to find all prime numbers up to a given number. Here is how it works for numbers 1 to 50:
Write all numbers from 1 to 50.
Cross out 1 (it is neither prime nor composite).
Circle 2 (first prime). Cross out all multiples of 2 (4, 6, 8, 10, ...).
Circle 3 (next uncrossed number = prime). Cross out all multiples of 3 (9, 15, 21, ...).
Circle 5 (next uncrossed). Cross out all multiples of 5 (25, 35, ...).
Circle 7 (next uncrossed). Cross out all multiples of 7 (49).
All remaining uncrossed numbers are prime!
Image needed: A 5×10 grid showing numbers 1-50. Number 1 is greyed out. Primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47) are highlighted in green circles. Composite numbers are crossed out in red. Title: "Sieve of Eratosthenes (1-50)" Style: Clean grid, colourful, child-friendly, clearly labelled Size: 700×400px
Quick Check — Prime or Composite?
Number
Factors
Type
13
1, 13
Prime (2 factors)
15
1, 3, 5, 15
Composite (4 factors)
23
1, 23
Prime (2 factors)
27
1, 3, 9, 27
Composite (4 factors)
29
1, 29
Prime (2 factors)
36
1, 2, 3, 4, 6, 9, 12, 18, 36
Composite (9 factors)
Prime Factorisation
Prime factorisation means breaking a number down into a product of prime numbers. Every composite number can be written as a product of prime factors in exactly one way (this is called the Fundamental Theorem of Arithmetic).
Method 1: Factor Tree
In a factor tree, we keep splitting the number into two factors until all factors at the bottom are prime numbers.
Example 1: Prime factorisation of 36 using a factor tree:
36
/ \
4 9
/ \ / \
2 2 3 3
36 = 2 × 2 × 3 × 3 = 2² × 3²
Example 2: Prime factorisation of 60:
60
/ \
6 10
/ \ / \
2 3 2 5
60 = 2 × 3 × 2 × 5 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Image needed: A colourful factor tree showing the prime factorisation of 72. The tree branches from 72 → 8 × 9 → (2×4)(3×3) → (2×2×2)(3×3). Prime factors at the leaves are circled in green. Title: "Factor Tree for 72" Style: Tree diagram with colourful branches, child-friendly, circles around primes Size: 600×450px
Example 3: Prime factorisation of 72:
72
/ \
8 9
/ \ / \
2 4 3 3
/ \
2 2
72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
Method 2: Division Method (Repeated Division by Primes)
In this method, we repeatedly divide the number by the smallest possible prime factor until we reach 1.
Example 4: Prime factorisation of 120 using division method:
2
| 120
2
| 60
2
| 30
3
| 15
5
| 5
| 1
120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
Example 5: Prime factorisation of 180:
2
| 180
2
| 90
3
| 45
3
| 15
5
| 5
| 1
180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5
Common Factors and HCF (Highest Common Factor)
Common factors are factors that appear in the factor lists of two or more numbers. The Highest Common Factor (HCF) is the largest of these common factors.
HCF is also called GCD (Greatest Common Divisor) in some textbooks.
Method 1: Listing Method
List all factors of each number, find the common ones, and pick the largest.
Example 1: Find the HCF of 12 and 18.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Image needed: A Venn diagram showing factors of 24 in the left circle and factors of 36 in the right circle. The overlapping region shows common factors (1, 2, 3, 4, 6, 12) with 12 highlighted as HCF. Title: "Finding HCF using Venn Diagram" Style: Two overlapping circles (blue and green), common region in yellow, HCF highlighted in red circle Size: 650×400px
Method 2: Prime Factorisation Method
To find HCF using prime factorisation:
Find the prime factorisation of each number.
Identify the common prime factors.
Multiply the common prime factors using their lowest powers.
Example 3: Find the HCF of 48 and 60 using prime factorisation.
Common prime factors: 2 and 3
Lowest powers: 2² and 3¹
HCF = 2² × 3 = 4 × 3 = 12
Example 4: Find the HCF of 48, 60, and 72.
48 = 2⁴ × 3
60 = 2² × 3 × 5
72 = 2³ × 3²
Common prime factors: 2 and 3
Lowest powers: 2² (lowest power of 2) and 3¹ (lowest power of 3)
HCF = 2² × 3 = 4 × 3 = 12
Think About It: The HCF of two numbers is always less than or equal to the smaller number. Why? Because a factor can never be greater than the number itself!
Common Multiples and LCM (Lowest Common Multiple)
Common multiples are multiples that appear in the multiple lists of two or more numbers. The Lowest Common Multiple (LCM) is the smallest of these common multiples.
Method 1: Listing Method
List multiples of each number until you find the first common one.
Common multiples: 60, 120, 180, ... LCM(12, 15) = 60
Image needed: A number line from 0 to 50 showing multiples of 6 (marked with blue dots above the line) and multiples of 8 (marked with red dots below the line). The point 24 is highlighted with a star where both meet — the LCM. Title: "Finding LCM on a Number Line" Style: Horizontal number line, colourful dots, child-friendly, clearly labelled Size: 800×300px
Method 2: Prime Factorisation Method
To find LCM using prime factorisation:
Find the prime factorisation of each number.
List all prime factors that appear in either number.
For each prime factor, use the highest power that appears.
Multiply them all together.
Example 3: Find the LCM of 12 and 15 using prime factorisation.
12 = 2² × 3
15 = 3 × 5
All prime factors: 2, 3, 5
Highest powers: 2², 3¹, 5¹
LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60
Example 4: Find the LCM of 4, 6, and 9.
4 = 2²
6 = 2 × 3
9 = 3²
All prime factors: 2, 3
Highest powers: 2² (from 4), 3² (from 9)
LCM = 2² × 3² = 4 × 9 = 36
Method 3: Common Division Method
This is a quick method where we divide all numbers together by the smallest prime that divides at least one of them, and continue until all quotients become 1.
Example 5: Find the LCM of 12, 18, and 24 using the common division method.
2
| 12
18
24
2
| 6
9
12
2
| 3
9
6
3
| 3
9
3
3
| 1
3
1
| 1
1
1
LCM = 2 × 2 × 2 × 3 × 3 = 72
Example 6: Find the LCM of 8, 10, and 15.
2
| 8
10
15
2
| 4
5
15
2
| 2
5
15
3
| 1
5
15
5
| 1
5
5
| 1
1
1
LCM = 2 × 2 × 2 × 3 × 5 = 120
Relationship Between HCF and LCM
For any two numbers, there is a beautiful relationship:
Important Formula
HCF × LCM = Product of the two numbers
That is: HCF(a, b) × LCM(a, b) = a × b
Note: This formula works only for two numbers, not three or more.
Using the Formula: The HCF of two numbers is 8 and their LCM is 96. If one number is 32, find the other number.
HCF × LCM = Product of two numbers
8 × 96 = 32 × other number
768 = 32 × other number
Other number = 768 ÷ 32 = 24
More Facts About HCF and LCM
HCF of two numbers is always a factor of their LCM.
LCM is always greater than or equal to the larger number.
HCF is always less than or equal to the smaller number.
If one number is a factor of the other, then HCF = smaller number and LCM = larger number.
If two numbers are co-prime (HCF = 1), then LCM = product of the two numbers.
Word Problems — HCF and LCM in Real Life
Here is how to decide whether a word problem needs HCF or LCM:
When to Use HCF vs LCM?
Use HCF when you need to divide things into equal groups, cut into equal pieces, or find the largest equal size.
Use LCM when you need to find when things happen together again, or find the smallest number that works for multiple groups.
HCF Problem 1: Sita has 42 red roses and 56 white roses. She wants to make bouquets with the same number of flowers in each bouquet, without mixing colours. What is the maximum number of flowers she can put in each bouquet?
We need the largest number that divides both 42 and 56 → find HCF.
42 = 2 × 3 × 7
56 = 2³ × 7
HCF = 2 × 7 = 14
She can put 14 flowers in each bouquet.
(42 ÷ 14 = 3 bouquets of red roses, 56 ÷ 14 = 4 bouquets of white roses)
HCF Problem 2: Ravi has a rectangular plot measuring 24 metres by 36 metres. He wants to divide it into the largest possible square plots. What is the side of each square?
Side of largest square = HCF(24, 36) = 12 metres
Number of squares = (24÷12) × (36÷12) = 2 × 3 = 6 squares
Image needed: An illustration showing two real-life HCF/LCM scenarios: (1) A girl cutting two ribbons of different lengths into equal pieces (HCF), (2) Two temple bells with a clock showing when they ring together (LCM). Split into two panels with labels. Style: Cartoon illustration, Indian context (girl in salwar kameez, temple bells), colourful, child-friendly Size: 750×400px
LCM Problem 1: A temple bell rings every 15 minutes and another bell rings every 20 minutes. Both bells rang together at 8:00 AM. When will they ring together again?
We need the smallest time after which both ring together → find LCM.
15 = 3 × 5
20 = 2² × 5
LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60 minutes
They will ring together again at 9:00 AM (60 minutes = 1 hour later).
LCM Problem 2: Arjun is preparing Diwali gift boxes. He has 16 laddoos, 24 barfis, and 40 chocolates. He wants to pack them in boxes so that each box has the same assortment and nothing is left over. What is the maximum number of boxes he can make?
We need the largest number that divides 16, 24, and 40 → find HCF.
16 = 2⁴
24 = 2³ × 3
40 = 2³ × 5
HCF = 2³ = 8 boxes
Each box will have: 16÷8 = 2 laddoos, 24÷8 = 3 barfis, 40÷8 = 5 chocolates.
LCM Problem 3: Meena goes swimming every 4 days, and Priya goes swimming every 6 days. They both went swimming today. After how many days will they go swimming on the same day again?
A number that divides another number exactly (no remainder)
Multiple
The result of multiplying a number by a whole number
Prime Number
A number with exactly 2 factors (1 and itself)
Composite Number
A number with more than 2 factors
Prime Factorisation
Writing a number as a product of prime numbers
Factor Tree
A diagram showing how a number splits into prime factors
Common Factors
Factors shared by two or more numbers
HCF
Highest Common Factor — the largest common factor of two or more numbers
Common Multiples
Multiples shared by two or more numbers
LCM
Lowest Common Multiple — the smallest common multiple of two or more numbers
Co-prime Numbers
Two numbers whose HCF is 1 (e.g., 8 and 15)
Key Points to Remember
Factors are finite and ≤ the number. Multiples are infinite and ≥ the number.
1 is a factor of every number. Every number is a multiple of 1 and itself.
2 is the only even prime number. All other even numbers are composite.
1 is neither prime nor composite.
To find HCF: multiply common prime factors with lowest powers.
To find LCM: multiply all prime factors with highest powers.
HCF × LCM = Product of two numbers (for exactly two numbers).
HCF is always ≤ the smaller number. LCM is always ≥ the larger number.
Use HCF for dividing/cutting into equal groups. Use LCM for finding when events coincide.
If two numbers are co-prime (HCF = 1), their LCM = their product.
Finished learning? Now test what you know!
🎯 Quick Quiz — Test Yourself!
Answer these questions right here. You get instant feedback — no paper needed!
Score: 0 / 8
1. Which of these is a prime number?
17 has only two factors: 1 and 17. The others are composite.
2. What is the HCF of 12 and 18?
Common factors of 12 and 18: 1, 2, 3, 6. Highest is 6.
3. What is the LCM of 6 and 8?
Multiples of 6: 6,12,18,24… Multiples of 8: 8,16,24… First common = 24.
4. Which number is a multiple of both 3 and 5?
15 = 3 × 5, so it is a multiple of both.
5. What is the HCF of two co-prime numbers (like 8 and 9)?
Co-prime numbers share only the factor 1, so their HCF is 1.
6. How many factors does 6 have?
Factors of 6: 1, 2, 3, 6 — that is 4 factors.
7. What is the LCM of 4 and 5?
4 and 5 are co-prime, so LCM = 4 × 5 = 20.
8. Two ribbons are 24 cm and 36 cm long. What is the longest equal length you can cut both into (HCF)?
HCF of 24 and 36 = 12 cm.
Practice Questions
A. Find all factors of the following numbers:
1. 28
2. 45
3. 56
4. 72
5. 100
B. Write the first 8 multiples of:
1. 7: , , , , , , ,
2. 9: , , , , , , ,
3. 11: , , , , , , ,
4. 15: , , , , , , ,
5. 20: , , , , , , ,
C. Write Prime (P) or Composite (C) for each number:
1. 17 =
2. 21 =
3. 31 =
4. 39 =
5. 43 =
6. 51 =
7. 67 =
8. 91 =
D. Find the prime factorisation using a factor tree:
1. 48
2. 90
3. 84
4. 150
E. Find the HCF of:
1. 16 and 24
2. 36 and 54
3. 42 and 63
4. 48, 72, and 108
5. 60 and 84
F. Find the LCM of:
1. 8 and 12
2. 15 and 20
3. 9 and 14
4. 6, 8, and 12
5. 10, 15, and 25
G. Fill in the blanks:
1. The smallest prime number is .
2. HCF of two co-prime numbers is always .
3. LCM of 5 and 7 = (since they are co-prime).
4. If HCF of two numbers is 6 and their LCM is 72, and one number is 24, the other number is .
5. The number of factors of a prime number is always .
H. True or False:
1. Every even number greater than 2 is composite.
2. HCF of two numbers can be greater than both numbers.
3. LCM of two numbers is always greater than both numbers.
4. 1 is a prime number.
5. The HCF of 12 and 36 is 12.
I. Choose the correct answer (MCQ):
1. The HCF of 18 and 27 is:
a) 3b) 6c) 9d) 18
2. The LCM of 4 and 6 is:
a) 2b) 12c) 24d) 6
3. Which of the following is NOT a prime number?
a) 37b) 41c) 49d) 53
4. If HCF(a, b) = a, then:
a) a is a multiple of bb) b is a multiple of ac) a = bd) a and b are co-prime
J. Word Problems:
1. Priya has two pieces of cloth — one 48 cm long and another 64 cm long. She wants to cut them into strips of equal length without wasting any cloth. What is the longest possible strip she can cut?
2. Two buses leave a bus stand at the same time. Bus A returns every 18 minutes, and Bus B returns every 24 minutes. After how many minutes will both buses be at the stand again at the same time?
3. Arjun wants to tile his bathroom floor (120 cm × 96 cm) with the largest possible square tiles. What is the side of each tile, and how many tiles will he need?
4. Ravi distributes 84 laddoos and 60 pedhas equally among some children. Each child gets the same number of laddoos and the same number of pedhas, with nothing left over. What is the maximum number of children?
5. Three traffic lights turn red at the same time. The first turns red every 36 seconds, the second every 48 seconds, and the third every 72 seconds. After how many seconds will all three turn red together again?
Activity — Factor Bingo!
Play this fun game with your classmates or family:
Setup: Each player draws a 4×4 grid and fills it with any 16 numbers from 1 to 50 (no repeats).
How to Play: The caller picks a number from 10 to 100 and announces it. All players cross out any numbers on their grid that are factors of the called number.
Example: If the caller says "36," you can cross out 1, 2, 3, 4, 6, 9, 12, 18, 36 (if any of these are on your grid).
Winning: The first player to cross out a complete row, column, or diagonal shouts "BINGO!" and wins!
Variation: Play "Multiple Bingo" — the caller says a small number (2-12), and you cross out any multiples of that number on your grid.
Fill this grid with your chosen numbers and start playing!
Want to review the lesson again?
Mathematics
HCF and LCM — Factors and Multiples
Class 5 | CBSE / NCERT / ICSE
Name: Class: Date:
Learning Objectives
Understand factors and multiples of a number
Identify prime and composite numbers
Find prime factorisation using factor tree and division method
Find HCF and LCM of two or more numbers
Apply HCF and LCM to solve word problems
Introduction
Priya has two ribbons — 24 cm and 36 cm. She wants to cut them into equal pieces with no waste. The longest piece = HCF. Arjun visits the library every 4 days, Meena every 6 days. When will they visit together again? That's the LCM! In Class 5, we learn HCF, LCM, and prime factorisation — powerful tools for everyday problems.
What Are Factors?
A factor divides a number exactly (remainder = 0). Example: 6 is a factor of 24 because 24 ÷ 6 = 4.
Properties: 1 is a factor of every number. Every number is a factor of itself. Factors are finite and ≤ the number.
Prime: Exactly 2 factors (1 and itself). E.g., 2, 3, 5, 7, 11, 13... Composite: More than 2 factors. E.g., 4, 6, 8, 9, 10... Special: 1 is neither prime nor composite. 2 is the only even prime.
More facts: HCF divides LCM. HCF ≤ smaller number. LCM ≥ larger number. If co-prime (HCF=1), LCM = product.
Word Problems — When to Use HCF vs LCM?
HCF: Divide/cut into equal groups, find largest equal size. LCM: Find when events happen together, smallest common quantity.
HCF: Sita has 42 roses and 56 white roses. Max flowers per bouquet without mixing? HCF(42,56) = 14 LCM: Bell A rings every 15 min, Bell B every 20 min. Ring together after? LCM(15,20) = 60 min
[Image: Two panels — (1) girl cutting ribbons into equal pieces (HCF), (2) temple bells ringing together (LCM)]
Key Words
Term
Meaning
Factor
Divides a number exactly
Multiple
Result of multiplying by a whole number
Prime
Exactly 2 factors (1 and itself)
Composite
More than 2 factors
HCF
Highest Common Factor (largest shared factor)
LCM
Lowest Common Multiple (smallest shared multiple)
Co-prime
Numbers with HCF = 1
Key Points:
• Factors: finite, ≤ number. Multiples: infinite, ≥ number.
• 2 is the only even prime. 1 is neither prime nor composite.
• HCF: common primes × lowest powers. LCM: all primes × highest powers.
• HCF × LCM = Product of two numbers.
• HCF → equal division. LCM → events coinciding.
Practice Questions
A. Find all factors of:
1. 28:
2. 45:
3. 56:
4. 72:
5. 100:
B. Write the first 8 multiples of:
1. 7: , , , , , , ,
2. 9: , , , , , , ,
3. 11: , , , , , , ,
4. 15: , , , , , , ,
5. 20: , , , , , , ,
C. Write Prime (P) or Composite (C):
1. 17 = 2. 21 = 3. 31 = 4. 39 =
5. 43 = 6. 51 = 7. 67 = 8. 91 =
D. Prime factorisation (draw factor tree):
1. 48
2. 90
3. 84
4. 150
E. Find the HCF of:
1. 16 and 24
2. 36 and 54
3. 42 and 63
4. 48, 72, and 108
5. 60 and 84
F. Find the LCM of:
1. 8 and 12
2. 15 and 20
3. 9 and 14
4. 6, 8, and 12
5. 10, 15, and 25
G. Fill in the blanks:
1. The smallest prime number is .
2. HCF of two co-prime numbers is always .
3. LCM of 5 and 7 = .
4. If HCF=6, LCM=72, one number=24, other = .
5. Number of factors of a prime number = .
H. True or False:
1. Every even number > 2 is composite.
2. HCF can be greater than both numbers.
3. LCM is always greater than both numbers.
4. 1 is a prime number.
5. HCF of 12 and 36 is 12.
I. MCQ:
1. HCF of 18 and 27 is:
a) 3b) 6c) 9d) 18
2. LCM of 4 and 6 is:
a) 2b) 12c) 24d) 6
3. Which is NOT prime?
a) 37b) 41c) 49d) 53
4. If HCF(a,b) = a, then:
a) a is multiple of bb) b is multiple of ac) a=bd) co-prime
J. Word Problems:
1. Priya has cloth pieces of 48 cm and 64 cm. Longest equal strip without waste?
2. Bus A returns every 18 min, Bus B every 24 min. When together again?
3. Floor 120cm × 96cm. Largest square tile side? How many tiles?
4. Ravi distributes 84 laddoos and 60 pedhas equally. Max children?
5. Lights turn red every 36s, 48s, 72s. All red together after?
Activity — Factor Bingo!
Fill the 4×4 grid with numbers 1-50. Caller picks a number (10-100). Cross out its factors on your grid. First row/column/diagonal complete = BINGO!
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