Common Multiples
Imagine two buses start from the same bus stop. Bus A comes every 6 minutes and Bus B comes every 8 minutes. Both buses are at the stop right now. When will both buses be at the stop at the same time again?
To answer this, you need to find the common multiples of 6 and 8. The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, ... and the multiples of 8 are 8, 16, 24, 32, 40, 48, ... The numbers that appear in both lists (24, 48, 72, ...) are the common multiples. So the buses will meet again after 24 minutes.
Common multiples are part of the Playing with Numbers chapter in Class 6 Maths. They lead directly to the concept of LCM (Least Common Multiple), which is one of the most important ideas in arithmetic.
What is Common Multiples - Grade 6 Maths (Playing with Numbers)?
Definition: A common multiple of two or more numbers is a number that is a multiple of each of those numbers.
Example:
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
- Common multiples of 3 and 4: 12, 24, 36, 48, 60, ...
Key facts about common multiples:
- The smallest common multiple is called the Least Common Multiple (LCM). In the example above, LCM of 3 and 4 = 12.
- Every common multiple is a multiple of the LCM. The common multiples of 3 and 4 are 12, 24, 36, 48, ... which are all multiples of 12.
- There are infinitely many common multiples of any two numbers.
- There is no greatest common multiple (the list never ends).
Common Multiples Formula
How to find common multiples:
Step 1: List multiples of each number.
Step 2: Pick the numbers that appear in ALL lists.
Shortcut using LCM:
- Find the LCM of the numbers.
- All common multiples are: LCM, 2 × LCM, 3 × LCM, 4 × LCM, ...
Example: Common multiples of 5 and 7:
- LCM of 5 and 7 = 35 (since 5 and 7 share no common factor)
- Common multiples: 35, 70, 105, 140, 175, ...
Relationship between LCM and common multiples:
Every common multiple = LCM × (some whole number)
Derivation and Proof
Step-by-step: Finding common multiples of 4, 6, and 8
- List multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, ...
- List multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...
- List multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, ...
- Common to all three: 24, 48, 72, 96, 120, ...
The LCM of 4, 6, and 8 is 24. Every common multiple is a multiple of 24.
Why does this work?
- If a number is divisible by 4, 6, and 8, it must contain all the prime factors of each.
- 4 = 2², 6 = 2 × 3, 8 = 2³
- LCM = 2³ × 3 = 24
- Every common multiple must be a multiple of 24 (because 24 is the smallest number containing all required prime factors).
Types and Properties
Types of common multiple problems:
Type 1: List the first few common multiples
- Find the first 4 common multiples of 5 and 6.
- Multiples of 5: 5, 10, 15, 20, 25, 30, 35, ..., 60, ..., 90, ..., 120, ...
- Multiples of 6: 6, 12, 18, 24, 30, 36, ..., 60, ..., 90, ..., 120, ...
- First 4 common multiples: 30, 60, 90, 120.
Type 2: Find the LCM (smallest common multiple)
- LCM is the smallest number in the common multiples list.
Type 3: Word problems about repeating events
- Two bells ring every 12 and 15 minutes. When do they ring together? Find LCM of 12 and 15 = 60. They ring together every 60 minutes.
Type 4: Common multiples of three or more numbers
- Find common multiples of 2, 3, and 5.
- LCM = 30. Common multiples: 30, 60, 90, 120, ...
Type 5: Checking if a number is a common multiple
- Is 120 a common multiple of 8 and 15? Check: 120÷8=15 (yes) and 120÷15=8 (yes). So yes.
Solved Examples
Example 1: Example 1: Common Multiples of Two Numbers
Problem: Find the first 5 common multiples of 3 and 5.
Solution:
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, ..., 60, ..., 75
- Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75
- Common multiples: 15, 30, 45, 60, 75
Answer: First 5 common multiples of 3 and 5 are 15, 30, 45, 60, 75.
Example 2: Example 2: Finding the LCM
Problem: Find the LCM of 6 and 9 using the listing method.
Solution:
- Multiples of 6: 6, 12, 18, 24, 30, 36, ...
- Multiples of 9: 9, 18, 27, 36, ...
- Common multiples: 18, 36, 54, ...
- The smallest (least) common multiple = 18
Answer: LCM of 6 and 9 = 18.
Example 3: Example 3: Common Multiples of Three Numbers
Problem: Find the first 3 common multiples of 2, 3, and 7.
Solution:
- LCM of 2, 3, and 7 = 2 × 3 × 7 = 42 (since all three are co-prime to each other)
- Common multiples: 42, 84, 126
Answer: First 3 common multiples of 2, 3, and 7 are 42, 84, 126.
Example 4: Example 4: Checking a Common Multiple
Problem: Is 72 a common multiple of 8 and 12?
Solution:
- 72 ÷ 8 = 9 (no remainder) → 72 is a multiple of 8. Yes.
- 72 ÷ 12 = 6 (no remainder) → 72 is a multiple of 12. Yes.
Answer: Yes, 72 is a common multiple of 8 and 12.
Example 5: Example 5: Bus Timing Problem
Problem: Bus A comes every 12 minutes and Bus B comes every 18 minutes. Both are at the stop at 9:00 AM. When will they next be at the stop together?
Solution:
- We need the LCM of 12 and 18.
- Multiples of 12: 12, 24, 36, 48, 60, 72, ...
- Multiples of 18: 18, 36, 54, 72, ...
- LCM = 36
- Both buses will be together after 36 minutes.
Answer: They will both be at the stop at 9:36 AM.
Example 6: Example 6: Bell Ringing Problem
Problem: Three bells ring at intervals of 4, 6, and 10 minutes. If they all ring together at 8:00 AM, when will they all ring together again?
Solution:
- Find LCM of 4, 6, and 10.
- 4 = 2², 6 = 2 × 3, 10 = 2 × 5
- LCM = 2² × 3 × 5 = 60
- They will ring together after 60 minutes = 1 hour.
Answer: They will all ring together again at 9:00 AM.
Example 7: Example 7: Smallest Number Divisible by Given Numbers
Problem: What is the smallest number that is divisible by both 8 and 14?
Solution:
- This is just the LCM of 8 and 14.
- Multiples of 8: 8, 16, 24, 32, 40, 48, 56, ...
- Multiples of 14: 14, 28, 42, 56, ...
- LCM = 56
Answer: The smallest number divisible by both 8 and 14 is 56.
Example 8: Example 8: Common Multiples and Division
Problem: Find the smallest number that when divided by 5, 6, and 8 leaves no remainder.
Solution:
- The number must be a common multiple of 5, 6, and 8.
- 5 = 5, 6 = 2 × 3, 8 = 2³
- LCM = 2³ × 3 × 5 = 8 × 15 = 120
Answer: The smallest such number is 120.
Example 9: Example 9: Cricket Training Schedule
Problem: Rohit practises batting every 3 days and bowling every 5 days. If he does both today, after how many days will he do both again on the same day?
Solution:
- We need the LCM of 3 and 5.
- Since 3 and 5 have no common factors, LCM = 3 × 5 = 15.
Answer: Rohit will do both batting and bowling on the same day after 15 days.
Example 10: Example 10: Common Multiples Less Than 100
Problem: Find all common multiples of 9 and 12 that are less than 100.
Solution:
- LCM of 9 and 12: 9 = 3², 12 = 2² × 3. LCM = 2² × 3² = 36.
- Common multiples: 36, 72, 108, ...
- Less than 100: 36 and 72
Answer: Common multiples of 9 and 12 less than 100 are 36 and 72.
Real-World Applications
Real-life uses of common multiples:
- Bus and train schedules: If two trains depart every 20 and 30 minutes, they depart together every 60 minutes (LCM of 20 and 30).
- Buying equal quantities: If pencils come in packs of 6 and erasers in packs of 8, you buy 24 of each (LCM of 6 and 8) to have equal numbers without leftovers.
- Tiling patterns: To tile a wall using tiles of two different sizes without cutting, the wall length should be a common multiple of both tile sizes.
- Sports practice: If a cricket coach conducts batting practice every 4 days and fielding every 6 days, combined sessions happen every 12 days.
- Adding fractions: To add 1/4 + 1/6, you need a common denominator, which is a common multiple of 4 and 6 (= 12). So 1/4 = 3/12 and 1/6 = 2/12.
- Traffic lights: If two traffic lights change every 45 and 60 seconds, they both turn green together every 180 seconds (LCM = 180).
Key Points to Remember
- A common multiple of two or more numbers is a number that is a multiple of each of them.
- The Least Common Multiple (LCM) is the smallest common multiple.
- Every common multiple is a multiple of the LCM.
- There are infinitely many common multiples of any set of numbers.
- To find common multiples by listing: write multiples of each number and find the ones that appear in all lists.
- The LCM of two co-prime numbers (no common factor except 1) equals their product.
- Common multiples are used to find common denominators when adding or subtracting fractions.
- If two events repeat at different intervals, they happen together at intervals equal to the LCM.
- LCM × HCF = Product of the two numbers (for any two numbers).
- The LCM of any number with itself is the number itself.
Practice Problems
- Find the first 4 common multiples of 4 and 7.
- Find the LCM of 8 and 12 using the listing method.
- Is 150 a common multiple of 15 and 25? Verify.
- Find the smallest number divisible by 3, 4, and 5.
- Two lights flash every 8 seconds and 12 seconds. If they flash together now, when will they flash together next?
- Find all common multiples of 6 and 8 that are less than 60.
- Find the LCM of 9, 12, and 15.
- Pens come in packs of 5 and notebooks in packs of 3. What is the minimum number of each you must buy to have equal numbers of pens and notebooks?
Frequently Asked Questions
Q1. What is a common multiple?
A common multiple of two or more numbers is a number that appears in the multiplication tables of all those numbers. For example, 24 is a common multiple of 3 and 8 because 3 × 8 = 24 and 8 × 3 = 24.
Q2. What is the LCM?
LCM stands for Least Common Multiple. It is the smallest positive common multiple of two or more numbers. For example, common multiples of 4 and 6 are 12, 24, 36, ... and the LCM is 12 (the smallest).
Q3. Is there a greatest common multiple?
No. Common multiples go on forever (infinitely). You can always find a bigger one by adding the LCM again. If the LCM of 4 and 6 is 12, the common multiples are 12, 24, 36, 48, ... — they never end.
Q4. What is the difference between common factors and common multiples?
Common factors are shared divisors — they are finite and never bigger than the smaller number. Common multiples are shared multiples — they are infinite and always bigger than or equal to both numbers. For 6 and 8: common factors are {1, 2} (finite), common multiples are {24, 48, 72, ...} (infinite).
Q5. How do I find the LCM of two numbers quickly?
Use prime factorisation. Write each number as a product of primes. Take the highest power of each prime. Multiply them. Example: LCM of 12 and 18. 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 4 × 9 = 36.
Q6. What is the LCM of two numbers that have no common factor?
If two numbers are co-prime (HCF = 1), their LCM equals their product. For example, LCM of 5 and 7 = 5 × 7 = 35, because 5 and 7 share no common factor.
Q7. Can the LCM be one of the numbers itself?
Yes. If one number is a multiple of the other, then the LCM is the larger number. For example, LCM of 4 and 12 = 12, because 12 is already a multiple of 4.
Q8. How are common multiples used in fractions?
When adding fractions with different denominators (like 1/3 + 1/4), you convert them to a common denominator. The common denominator is a common multiple of 3 and 4. The LCM (12) is the best choice: 1/3 = 4/12, 1/4 = 3/12, so 1/3 + 1/4 = 7/12.
Q9. What is the formula connecting HCF and LCM?
For any two numbers a and b: HCF(a, b) × LCM(a, b) = a × b. For example, for 12 and 18: HCF = 6, LCM = 36. Check: 6 × 36 = 216 = 12 × 18. Correct.
Q10. What is the LCM of a number with itself?
The LCM of any number with itself is the number itself. LCM(7, 7) = 7. This makes sense because 7 is already a multiple of 7.
