Common Multiples
Common multiples are multiples that two or more numbers share. Finding common multiples is an important skill that leads to understanding the Least Common Multiple (LCM).
In Class 4, students learn to list multiples of two numbers, identify the shared ones, and apply this knowledge to solve problems about repeating events and scheduling.
What is Common Multiples - Class 4 Maths (Factors and Multiples)?
A common multiple of two or more numbers is a number that is a multiple of each of them.
For example, multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Common multiples of 4 and 6: 12, 24, 36, 48, ...
Common multiples of A and B = numbers that are divisible by both A and B
Common Multiples Formula
Steps to find common multiples:
- List several multiples of the first number.
- List several multiples of the second number.
- Identify numbers that appear in both lists.
Key fact: The smallest common multiple (other than 0) is called the Least Common Multiple (LCM).
Types and Properties
Important properties of common multiples:
- Common multiples are infinite — there is no largest common multiple.
- The smallest common multiple (LCM) is the most important one.
- Every other common multiple is a multiple of the LCM. For example, LCM of 4 and 6 is 12. Common multiples: 12, 24, 36, 48, ... (all multiples of 12).
- If one number is a multiple of the other, the LCM is the larger number. Example: LCM of 3 and 9 is 9.
Solved Examples
Example 1: Example 1: Common Multiples of 3 and 4
Problem: Find the first 4 common multiples of 3 and 4.
Solution:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48
Answer: First 4 common multiples of 3 and 4 = 12, 24, 36, 48
Example 2: Example 2: Common Multiples of 5 and 6
Problem: Find the first 3 common multiples of 5 and 6.
Solution:
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90
Answer: First 3 common multiples = 30, 60, 90
Example 3: Example 3: When One Divides the Other
Problem: Find the common multiples of 4 and 8.
Solution:
Since 8 is a multiple of 4, every multiple of 8 is automatically a multiple of 4.
Common multiples = multiples of 8 = 8, 16, 24, 32, 40, ...
Answer: Common multiples of 4 and 8 = 8, 16, 24, 32, 40, ... (all multiples of 8)
Example 4: Example 4: Finding the LCM
Problem: Find the Least Common Multiple (LCM) of 6 and 8.
Solution:
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48
Multiples of 8: 8, 16, 24, 32, 40, 48
The smallest common multiple is 24.
Answer: LCM of 6 and 8 = 24
Example 5: Example 5: Word Problem (Bells Ringing)
Problem: In a school, one bell rings every 4 minutes and another rings every 6 minutes. Both ring together at 9:00 AM. When will they ring together again?
Solution:
Find the LCM of 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20, 24
Multiples of 6: 6, 12, 18, 24
LCM = 12 minutes
Answer: They will ring together again at 9:12 AM (after 12 minutes).
Example 6: Example 6: Word Problem (Bus Schedule)
Problem: Bus A comes every 5 minutes. Bus B comes every 8 minutes. Both arrive at a stop at 8:00 AM. When will both arrive together again?
Solution:
Find the LCM of 5 and 8.
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40
Multiples of 8: 8, 16, 24, 32, 40
LCM = 40 minutes
Answer: Both buses arrive together again at 8:40 AM.
Example 7: Example 7: Is a Number a Common Multiple?
Problem: Is 60 a common multiple of 4 and 5?
Solution:
60 ÷ 4 = 15 (exact) ✓
60 ÷ 5 = 12 (exact) ✓
Since 60 is divisible by both 4 and 5, it is a common multiple.
Answer: Yes, 60 is a common multiple of 4 and 5.
Example 8: Example 8: Common Multiples of Three Numbers
Problem: Find the first 2 common multiples of 2, 3, and 5.
Solution:
A common multiple of 2, 3, and 5 must be divisible by all three.
LCM of 2, 3, 5 = 30 (since 2, 3, 5 are all prime).
Common multiples = multiples of 30: 30, 60, 90, ...
Answer: First 2 common multiples = 30, 60
Example 9: Example 9: All Common Multiples Are Multiples of LCM
Problem: The LCM of 6 and 9 is 18. List the first 5 common multiples.
Solution:
Since every common multiple is a multiple of the LCM:
18 × 1 = 18, 18 × 2 = 36, 18 × 3 = 54, 18 × 4 = 72, 18 × 5 = 90
Answer: First 5 common multiples = 18, 36, 54, 72, 90
Example 10: Example 10: Word Problem (Tiling)
Problem: Arjun has tiles that are 3 cm long and tiles that are 5 cm long. He wants to make two rows of tiles (one of each type) of equal length. What is the shortest length he can make?
Solution:
The length must be a common multiple of 3 and 5.
LCM of 3 and 5 = 15
Row of 3-cm tiles: 5 tiles (5 × 3 = 15)
Row of 5-cm tiles: 3 tiles (3 × 5 = 15)
Answer: The shortest equal length is 15 cm.
Real-World Applications
Common multiples are used in real life for:
- Scheduling: Finding when two events coincide (bus schedules, bell timings).
- Fractions: Finding a common denominator to add or compare fractions.
- Tiling/patterns: Finding lengths where different-sized tiles align.
- Packaging: Matching quantities from different pack sizes.
Key Points to Remember
- A common multiple is a number that is a multiple of two or more given numbers.
- Common multiples are infinite — they never end.
- The smallest common multiple (excluding 0) is the LCM.
- Every common multiple is a multiple of the LCM.
- If one number divides the other, the LCM is the larger number.
- To check if a number is a common multiple, verify it is divisible by all the given numbers.
- Common multiples are used to find common denominators for fractions.
Practice Problems
- Find the first 3 common multiples of 4 and 5.
- What is the LCM of 3 and 7?
- Is 48 a common multiple of 6 and 8? How do you check?
- Find the first 3 common multiples of 2, 3, and 4.
- A flashing light blinks every 3 seconds. Another blinks every 5 seconds. They both blink together. After how many seconds will they blink together again?
- Priya wants to buy the same number of apples and oranges. Apples come in packs of 6, oranges in packs of 8. What is the smallest number she can buy of each?
- Find the LCM of 9 and 12.
- Kavi says the common multiples of 4 and 6 are 12, 24, 36, 48, ... Is he correct? What pattern do these numbers follow?
Frequently Asked Questions
Q1. What is the difference between common factors and common multiples?
Common factors are shared factors (numbers that divide both). They are finite and small. Common multiples are shared multiples (numbers both divide into). They are infinite and larger.
Q2. What is the LCM?
The Least Common Multiple (LCM) is the smallest positive number that is a multiple of all given numbers. For 4 and 6, the LCM is 12.
Q3. Do common multiples ever end?
No. Common multiples are infinite. If m is a common multiple, then 2m, 3m, 4m, ... are also common multiples.
Q4. How are common multiples related to the LCM?
Every common multiple is a multiple of the LCM. Once you find the LCM, all common multiples are: LCM × 1, LCM × 2, LCM × 3, ...
Q5. What if one number is a multiple of the other?
Then the LCM is simply the larger number. For example, LCM of 5 and 15 is 15, because 15 is already a multiple of 5.
Q6. How do common multiples help with fractions?
To add fractions with different denominators (like 1/4 + 1/6), find a common multiple of 4 and 6 to use as the common denominator. LCM of 4 and 6 is 12, so convert to 3/12 + 2/12 = 5/12.
Q7. Can I find common multiples of three numbers?
Yes. The common multiples of three numbers must be divisible by all three. Find the LCM of all three, then list its multiples.
Q8. Is this topic in the NCERT Class 4 syllabus?
Yes. NCERT Class 4 Maths introduces common multiples as part of the factors and multiples chapter. LCM is explored further in Class 5.
Q9. Is 0 a common multiple?
Technically yes (0 is a multiple of every number), but we usually exclude 0 and look for positive common multiples. The LCM is always a positive number.
Related Topics
- Multiples of a Number
- LCM (Least Common Multiple) Introduction
- Factors of a Number
- Common Factors
- HCF (Highest Common Factor) Introduction
- Prime and Composite Numbers
- Sieve of Eratosthenes
- Even and Odd Number Properties
- Factors and Multiples Word Problems
- Prime Numbers up to 100
- Factor Pairs
- Patterns in Multiples
