LCM (Least Common Multiple) Introduction
The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of those numbers. Finding the LCM helps when you need to find common timing, combine groups, or work with fractions that have different denominators.
In Class 4, you will learn to find the LCM using the listing method and the prime factorisation method.
What is LCM (Least Common Multiple) Introduction - Class 4 Maths (Factors and Multiples)?
The Least Common Multiple (LCM) of two or more numbers is the smallest number (other than 0) that is a multiple of all the given numbers.
For example, multiples of 4 are: 4, 8, 12, 16, 20, 24, ... Multiples of 6 are: 6, 12, 18, 24, 30, ... The common multiples are 12, 24, 36, ... The smallest is 12. So, LCM of 4 and 6 = 12.
LCM (Least Common Multiple) Introduction Formula
LCM = The smallest number in the list of common multiples
Steps to find LCM by listing multiples:
- Write the first several multiples of each number.
- Find the multiples that appear in every list.
- The smallest common multiple is the LCM.
Types and Properties
Methods to Find LCM:
- Listing Method: Write multiples of each number until you find a common one. The smallest common multiple is the LCM.
- Prime Factorisation Method: Write each number as a product of prime factors. Take each prime factor the greatest number of times it appears. Multiply them together.
Prime Factorisation Example:
Find LCM of 12 and 18:
12 = 2 × 2 × 3
18 = 2 × 3 × 3
Take each prime the most times it appears: 2 × 2 × 3 × 3 = 36
LCM of 12 and 18 = 36
Solved Examples
Example 1: Example 1: LCM by Listing Multiples
Problem: Find the LCM of 3 and 5.
Solution:
Step 1: Multiples of 3 = 3, 6, 9, 12, 15, 18, ...
Step 2: Multiples of 5 = 5, 10, 15, 20, 25, ...
Step 3: The smallest common multiple = 15
Answer: LCM of 3 and 5 = 15
Example 2: Example 2: LCM by Listing Multiples
Problem: Find the LCM of 6 and 8.
Solution:
Step 1: Multiples of 6 = 6, 12, 18, 24, 30, ...
Step 2: Multiples of 8 = 8, 16, 24, 32, 40, ...
Step 3: The smallest common multiple = 24
Answer: LCM of 6 and 8 = 24
Example 3: Example 3: LCM by Prime Factorisation
Problem: Find the LCM of 10 and 15 using prime factorisation.
Solution:
Step 1: 10 = 2 × 5
Step 2: 15 = 3 × 5
Step 3: Take each prime factor the most times it appears: 2 × 3 × 5 = 30
Answer: LCM of 10 and 15 = 30
Example 4: Example 4: LCM When One Number is a Multiple of the Other
Problem: Find the LCM of 4 and 20.
Solution:
Step 1: Multiples of 4 = 4, 8, 12, 16, 20, 24, ...
Step 2: Multiples of 20 = 20, 40, 60, ...
Step 3: The smallest common multiple = 20
Answer: LCM of 4 and 20 = 20
When one number is a multiple of the other, the larger number is the LCM.
Example 5: Example 5: LCM of Co-prime Numbers
Problem: Find the LCM of 7 and 9.
Solution:
Step 1: 7 and 9 are co-prime (their HCF is 1).
Step 2: For co-prime numbers, LCM = product of the numbers.
Step 3: LCM = 7 × 9 = 63
Answer: LCM of 7 and 9 = 63
Example 6: Example 6: LCM of Three Numbers
Problem: Find the LCM of 2, 3, and 5.
Solution:
Step 1: Multiples of 2 = 2, 4, 6, ..., 30, ...
Step 2: Multiples of 3 = 3, 6, 9, ..., 30, ...
Step 3: Multiples of 5 = 5, 10, 15, 20, 25, 30, ...
Step 4: The smallest common multiple = 30
Answer: LCM of 2, 3, and 5 = 30
Example 7: Example 7: Word Problem
Problem: Ria goes to the library every 4 days and Kavi goes every 6 days. They both went today. After how many days will they go to the library on the same day again?
Solution:
Step 1: We need the LCM of 4 and 6.
Step 2: Multiples of 4 = 4, 8, 12, 16, 20, ...
Step 3: Multiples of 6 = 6, 12, 18, 24, ...
Step 4: LCM = 12
Answer: They will go to the library on the same day after 12 days.
Example 8: Example 8: Word Problem
Problem: Dev has two bells. One rings every 5 minutes and the other rings every 8 minutes. Both bells ring together at 9:00 AM. When will they ring together again?
Solution:
Step 1: Find LCM of 5 and 8.
Step 2: 5 = 5 (prime number)
Step 3: 8 = 2 × 2 × 2
Step 4: LCM = 2 × 2 × 2 × 5 = 40
Step 5: 40 minutes after 9:00 AM = 9:40 AM
Answer: Both bells will ring together again at 9:40 AM.
Example 9: Example 9: LCM by Prime Factorisation (Larger Numbers)
Problem: Find the LCM of 16 and 24 using prime factorisation.
Solution:
Step 1: 16 = 2 × 2 × 2 × 2
Step 2: 24 = 2 × 2 × 2 × 3
Step 3: Take each prime the most times it appears: 2 appears 4 times (from 16), 3 appears 1 time (from 24).
Step 4: LCM = 2 × 2 × 2 × 2 × 3 = 48
Answer: LCM of 16 and 24 = 48
Real-World Applications
Where do we use LCM?
- Finding common timing: When two events repeat at different intervals, use LCM to find when they happen together.
- Adding fractions: To add fractions with different denominators, find the LCM of the denominators to get the LCD.
- Planning schedules: If buses arrive every 10 minutes and 15 minutes, LCM tells when both arrive at the same time.
- Buying items in bulk: To buy items that come in packs of different sizes with no leftovers, use LCM.
Key Points to Remember
- The LCM of two or more numbers is the smallest common multiple of all of them.
- LCM is always greater than or equal to the largest of the given numbers.
- If one number is a multiple of the other, the larger number is the LCM.
- For co-prime numbers, LCM = product of the numbers.
- LCM can be found by listing multiples or by prime factorisation.
- HCF × LCM = Product of the two numbers (for two numbers only).
- LCM is used in adding fractions with different denominators.
Practice Problems
- Find the LCM of 4 and 10.
- Find the LCM of 9 and 12.
- Find the LCM of 8 and 14 using prime factorisation.
- Find the LCM of 3, 4, and 6.
- Aditi waters her plants every 3 days and Neha waters hers every 5 days. They both watered today. After how many days will they water on the same day?
- Find the LCM of 11 and 13. What do you notice about these numbers?
- Two traffic lights blink every 6 seconds and 9 seconds. They blink together now. After how many seconds will they blink together again?
Frequently Asked Questions
Q1. What is LCM in maths?
LCM stands for Least Common Multiple. It is the smallest number that is a multiple of two or more given numbers. For example, the LCM of 4 and 6 is 12.
Q2. What is the difference between LCM and HCF?
LCM is the smallest common multiple of the given numbers, while HCF is the largest common factor. LCM is always greater than or equal to the numbers, and HCF is always less than or equal to them.
Q3. How do you find LCM by listing multiples?
Write the multiples of each number until you find a number that appears in all lists. The first (smallest) number that is common to all lists is the LCM.
Q4. What is the LCM of two co-prime numbers?
For co-prime numbers (numbers whose HCF is 1), the LCM is simply the product of the two numbers. For example, LCM of 7 and 11 = 7 x 11 = 77.
Q5. Can LCM be smaller than the given numbers?
No. The LCM is always greater than or equal to the largest of the given numbers. It equals the largest number only when that number is a multiple of all the others.
Q6. What is the relationship between HCF and LCM?
For any two numbers, HCF x LCM = Product of the two numbers. For example, HCF(4, 6) = 2 and LCM(4, 6) = 12. Check: 2 x 12 = 24 = 4 x 6.
Q7. How is LCM used in fractions?
When adding or subtracting fractions with different denominators, you find the LCM of the denominators. This becomes the Least Common Denominator (LCD), which you use to convert the fractions before adding or subtracting.
Q8. What is the LCM of a number and 1?
The LCM of any number and 1 is the number itself. Since every number is a multiple of 1, the smallest common multiple is the number itself. For example, LCM(8, 1) = 8.
Q9. Is LCM covered in the NCERT Class 4 syllabus?
Yes. LCM is introduced in Class 4 under the chapter Factors and Multiples. Students learn the listing method and prime factorisation method to find the LCM.
