LCM by Prime Factorisation
The LCM (Least Common Multiple) of two or more numbers is the smallest number that is a multiple of all of them. It is also called the Lowest Common Multiple.
For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that appears in the multiples of both 4 (4, 8, 12, 16, ...) and 6 (6, 12, 18, 24, ...).
In Class 5, students learn to find the LCM using the prime factorisation method. This is the most reliable and efficient method for larger numbers.
What is LCM by Prime Factorisation - Class 5 Maths (Factors and Multiples)?
LCM (Least Common Multiple): The smallest positive number that is a multiple of two or more given numbers.
Finding LCM by prime factorisation:
- Find the prime factorisation of each number.
- List all prime factors that appear in any of the numbers.
- For each prime factor, take the one with the highest power.
- Multiply these together. The product is the LCM.
LCM = Product of all prime factors with the highest powers
Important properties of LCM:
- LCM is always greater than or equal to the largest of the given numbers.
- If one number is a multiple of another, the LCM is the larger number.
- LCM of two co-prime numbers is their product: LCM(7, 9) = 63.
- LCM of a number and itself is the number: LCM(15, 15) = 15.
LCM by Prime Factorisation Formula
LCM = Product of all prime factors (each raised to the highest power)
Useful relationship:
HCF(a, b) x LCM(a, b) = a x b
So: LCM(a, b) = (a x b) / HCF(a, b). This shortcut is very useful when you already know the HCF.
Types and Properties
Step-by-step method:
Example: Find LCM of 12 and 18
| Step | 12 | 18 |
|---|---|---|
| Prime factorisation | 2 x 2 x 3 | 2 x 3 x 3 |
| With exponents | 2² x 3¹ | 2¹ x 3² |
| All prime factors | 2 and 3 | |
| Highest power | 2² (higher of 2² and 2¹) and 3² (higher of 3¹ and 3²) | |
| LCM | 2² x 3² = 4 x 9 = 36 | |
HCF vs LCM comparison:
| Feature | HCF | LCM |
|---|---|---|
| Stands for | Highest Common Factor | Least Common Multiple |
| Uses | Common factors with LOWEST powers | ALL factors with HIGHEST powers |
| Result is | ≤ smallest number | ≥ largest number |
| Used for | Cutting, dividing, simplifying | Scheduling, repeating events, adding fractions |
Solved Examples
Example 1: Example 1: LCM of Two Numbers
Problem: Find the LCM of 12 and 18.
Solution:
Step 1: 12 = 2² x 3
Step 2: 18 = 2 x 3²
Step 3: All prime factors: 2 and 3
Step 4: Highest powers: 2² and 3²
Step 5: LCM = 2² x 3² = 4 x 9 = 36
Answer: LCM of 12 and 18 = 36
Example 2: Example 2: LCM of Larger Numbers
Problem: Find the LCM of 48 and 60.
Solution:
Step 1: 48 = 2⁴ x 3
Step 2: 60 = 2² x 3 x 5
Step 3: All prime factors: 2, 3, and 5
Step 4: Highest powers: 2⁴, 3¹, 5¹
Step 5: LCM = 16 x 3 x 5 = 240
Answer: LCM of 48 and 60 = 240
Example 3: Example 3: LCM of Three Numbers
Problem: Find the LCM of 8, 12, and 15.
Solution:
Step 1: 8 = 2³
Step 2: 12 = 2² x 3
Step 3: 15 = 3 x 5
Step 4: All prime factors: 2, 3, 5
Step 5: Highest powers: 2³, 3¹, 5¹
Step 6: LCM = 8 x 3 x 5 = 120
Answer: LCM of 8, 12, and 15 = 120
Example 4: Example 4: LCM of Co-Prime Numbers
Problem: Find the LCM of 7 and 11.
Solution:
Step 1: 7 = 7 (prime)
Step 2: 11 = 11 (prime)
Step 3: No common prime factors. All factors: 7 and 11.
Step 4: LCM = 7 x 11 = 77
Answer: LCM of 7 and 11 = 77 (for co-prime numbers, LCM = product)
Example 5: Example 5: When One Number is a Multiple of the Other
Problem: Find the LCM of 15 and 45.
Solution:
Step 1: 15 = 3 x 5
Step 2: 45 = 3² x 5
Step 3: Highest powers: 3² and 5¹
Step 4: LCM = 9 x 5 = 45
Answer: LCM of 15 and 45 = 45 (since 45 is a multiple of 15, the LCM is 45 itself)
Example 6: Example 6: Word Problem — Bell Ringing
Problem: In a temple, one bell rings every 15 minutes and another rings every 20 minutes. Both bells ring together at 6:00 AM. When will they ring together again?
Solution:
Step 1: We need the LCM of 15 and 20.
Step 2: 15 = 3 x 5
Step 3: 20 = 2² x 5
Step 4: LCM = 2² x 3 x 5 = 4 x 3 x 5 = 60 minutes
Step 5: 60 minutes = 1 hour after 6:00 AM = 7:00 AM
Answer: They will ring together again at 7:00 AM.
Example 7: Example 7: Word Problem — Stacking Boxes
Problem: Dev has boxes that are 6 cm tall, 8 cm tall, and 10 cm tall. He stacks each type separately to make equal-height towers. What is the shortest height at which all three towers will be the same?
Solution:
Step 1: We need the LCM of 6, 8, and 10.
Step 2: 6 = 2 x 3, 8 = 2³, 10 = 2 x 5
Step 3: All factors: 2, 3, 5. Highest powers: 2³, 3¹, 5¹
Step 4: LCM = 8 x 3 x 5 = 120 cm
Answer: The shortest equal height is 120 cm. (Dev needs 20 boxes of 6 cm, 15 boxes of 8 cm, and 12 boxes of 10 cm.)
Example 8: Example 8: Word Problem — Bus Timings
Problem: Bus A leaves the station every 12 minutes and Bus B every 18 minutes. Both leave at 8:00 AM. When will both buses leave together next?
Solution:
Step 1: LCM of 12 and 18
Step 2: 12 = 2² x 3, 18 = 2 x 3²
Step 3: LCM = 2² x 3² = 4 x 9 = 36 minutes
Step 4: 8:00 AM + 36 minutes = 8:36 AM
Answer: Both buses leave together next at 8:36 AM.
Example 9: Example 9: Using HCF-LCM Relationship
Problem: The HCF of two numbers is 4. One number is 28 and the other is 36. Find their LCM.
Solution:
Step 1: HCF x LCM = Product of the numbers
Step 2: 4 x LCM = 28 x 36 = 1,008
Step 3: LCM = 1,008 / 4 = 252
Answer: LCM of 28 and 36 = 252
Example 10: Example 10: Adding Fractions Using LCM
Problem: Add 5/12 + 7/18. Use LCM to find the common denominator.
Solution:
Step 1: LCM of 12 and 18 = 36 (from Example 1)
Step 2: Convert fractions: 5/12 = 15/36 (multiply by 3), 7/18 = 14/36 (multiply by 2)
Step 3: Add: 15/36 + 14/36 = 29/36
Answer: 5/12 + 7/18 = 29/36
Real-World Applications
Real-life uses of LCM:
- Scheduling: Finding when two recurring events happen at the same time (bus arrivals, bell ringing, shift rotations).
- Adding fractions: The LCM of the denominators gives the LCD (Least Common Denominator) needed to add or subtract fractions.
- Stacking and packing: Finding the smallest equal measurement from different-sized items.
- Circular tracks: Two runners on a circular track starting together — LCM of their lap times tells when they meet at the start again.
- Tile patterns: Finding when a repeating pattern of different lengths aligns.
Key Points to Remember
- LCM = Least Common Multiple = smallest number that is a multiple of all given numbers.
- To find LCM by prime factorisation: take all prime factors with the highest powers.
- LCM is always greater than or equal to the largest of the given numbers.
- For co-prime numbers, LCM = product of the numbers.
- If one number is a multiple of another, the LCM is the larger number.
- HCF x LCM = Product of the two numbers (for any two numbers).
- LCM is used to find the LCD (Least Common Denominator) when adding fractions.
- HCF uses lowest powers of common factors; LCM uses highest powers of all factors.
Practice Problems
- Find the LCM of 16 and 24 using prime factorisation.
- Find the LCM of 36 and 54.
- Find the LCM of 6, 10, and 15.
- Two lights flash every 8 seconds and 14 seconds. They flash together. After how many seconds will they flash together again?
- Find the LCD (LCM of denominators) to add 3/8 + 5/12.
- The LCM of two numbers is 180 and their HCF is 6. If one number is 30, find the other.
- Three runners take 12, 15, and 20 minutes to complete a lap. If they start together, after how many minutes will they all be at the start line again?
- Find the smallest number that is divisible by both 18 and 24.
Frequently Asked Questions
Q1. What is LCM?
LCM stands for Least Common Multiple. It is the smallest positive number that is a multiple of two or more given numbers. For example, the LCM of 6 and 8 is 24, because 24 is the smallest number divisible by both 6 and 8.
Q2. How do you find LCM using prime factorisation?
Find the prime factorisation of each number. List all prime factors that appear in any number. For each prime factor, take the highest power. Multiply them together. For example, LCM of 12 (2 squared x 3) and 18 (2 x 3 squared) = 2 squared x 3 squared = 36.
Q3. What is the difference between HCF and LCM?
HCF uses common prime factors with the lowest powers and gives a result smaller than or equal to the smallest number. LCM uses all prime factors with the highest powers and gives a result larger than or equal to the largest number. HCF divides, LCM multiplies.
Q4. Can LCM be smaller than the given numbers?
No. The LCM is always greater than or equal to the largest of the given numbers. The smallest possible LCM equals the largest number (when it is a multiple of all the others).
Q5. What is the LCM of co-prime numbers?
For co-prime numbers (whose HCF is 1), the LCM is simply their product. For example, LCM of 5 and 9 is 5 x 9 = 45, since they share no common factors.
Q6. How is LCM used in adding fractions?
To add fractions with different denominators, find the LCM of the denominators (called the LCD). Convert each fraction to an equivalent fraction with the LCD as the denominator. Then add the numerators. For 1/4 + 1/6: LCD = LCM(4,6) = 12, so 3/12 + 2/12 = 5/12.
Q7. What is the HCF-LCM relationship formula?
For any two numbers a and b: HCF(a, b) x LCM(a, b) = a x b. This means LCM = (a x b) / HCF. This shortcut avoids redoing prime factorisation if you already know the HCF.
Q8. Can LCM be found for more than two numbers?
Yes. Find the prime factorisation of all numbers. Take every prime factor that appears in any number, using the highest power. Multiply them. Or find the LCM of the first two, then find the LCM of that result with the third number.
Q9. 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. For example, LCM(7, 1) = 7.
Q10. When do you use LCM in real life?
LCM is used whenever you need to find when repeating events coincide. For example, if two buses come every 10 and 15 minutes, LCM(10, 15) = 30 tells you they arrive together every 30 minutes. It is also used for adding fractions with different denominators.
