Understanding L.C.M and H.C.F. Using Prime Factorization Method
Learning L.C.M and H.C.F. Using Prime Factorization Method helps students solve problems step by step. This topic is explained clearly in Mathematics Study Material for Hearing Impaired Students. Teachers often include counting lessons for hearing impaired students and math activities for hearing impaired children to make concepts interactive. As a result, learners build confidence through easy math study material for hearing impaired and inclusive math education for hearing impaired.
Study Material and Learning Resources
The best study material for hearing impaired students includes worksheets, activities, and examples using Prime Factorization. Math worksheets for hearing impaired students and a study guide for hearing impaired learners provide practice. Furthermore, special education math resources for hearing impaired support teachers in teaching numbers to hearing impaired students. With these math learning resources for hearing impaired, every child can master the process of L.C.M and H.C.F easily.
LCM and HCF using Prime Factorization
When working with two or more numbers, we often need to find their LCM (Least Common Multiple) and HCF (Highest Common Factor), also known as GCD (Greatest Common Divisor). One of the most systematic methods to find both is through Prime Factorization.
Prime Factorization Example:
36 = 2 × 2 × 3 × 3 = 2² × 3²
60 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹
1 Prime Factorization
Break down each number into its prime factors (prime numbers that multiply together to make the original number). Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves (2, 3, 5, 7, 11, 13, etc.).
Example: Find the prime factors of 36 and 60
In exponential form: 36 = 2² × 3², 60 = 2² × 3¹ × 5¹
2 HCF (Highest Common Factor)
To find the HCF, identify the common prime factors and take the lowest power of each common factor.
Example: Find HCF of 36 and 60
Common factors: 2² and 3¹ (5 is not common to both numbers)
HCF = 2² × 3¹ = 4 × 3 = 12
3 LCM (Least Common Multiple)
To find the LCM, take all prime factors from both numbers and use the highest power of each factor.
Example: Find LCM of 36 and 60
All factors: 2² (highest), 3² (highest), and 5¹
LCM = 2² × 3² × 5¹ = 4 × 9 × 5 = 180
4 Verification
Always verify your answer using the relationship:
LCM × HCF = Product of the original numbers
Example: Verify LCM and HCF of 36 and 60
LCM = 180, HCF = 12
180 × 12 = 2160
36 × 60 = 2160
✅ The relationship holds true!
Solved Examples
Example 1: Find LCM and HCF of 18 and 24
Prime factorization:
18 = 2 × 3 × 3 = 2¹ × 3²
24 = 2 × 2 × 2 × 3 = 2³ × 3¹
HCF: Common factors: 2¹ and 3¹ = 2 × 3 = 6
LCM: All factors: 2³ and 3² = 8 × 9 = 72
Verification: 6 × 72 = 432, 18 × 24 = 432 ✅
Example 2: Find LCM and HCF of 12 and 30
Prime factorization:
12 = 2 × 2 × 3 = 2² × 3¹
30 = 2 × 3 × 5 = 2¹ × 3¹ × 5¹
HCF: Common factors: 2¹ and 3¹ = 2 × 3 = 6
LCM: All factors: 2², 3¹, 5¹ = 4 × 3 × 5 = 60
Verification: 6 × 60 = 360, 12 × 30 = 360 ✅
Example 3: Find LCM and HCF of 15 and 25
Prime factorization:
15 = 3 × 5 = 3¹ × 5¹
25 = 5 × 5 = 5²
HCF: Common factors: 5¹ = 5
LCM: All factors: 3¹ and 5² = 3 × 25 = 75
Verification: 5 × 75 = 375, 15 × 25 = 375 ✅
Example 4: Find LCM and HCF of 16 and 28
Prime factorization:
16 = 2 × 2 × 2 × 2 = 2⁴
28 = 2 × 2 × 7 = 2² × 7¹
HCF: Common factors: 2² = 4
LCM: All factors: 2⁴ and 7¹ = 16 × 7 = 112
Verification: 4 × 112 = 448, 16 × 28 = 448 ✅
Example 5: Find LCM and HCF of 45 and 120
Prime factorization:
45 = 3 × 3 × 5 = 3² × 5¹
120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3¹ × 5¹
HCF: Common factors: 3¹ and 5¹ = 3 × 5 = 15
LCM: All factors: 2³, 3², 5¹ = 8 × 9 × 5 = 360
Verification: 15 × 360 = 5400, 45 × 120 = 5400 ✅
Practice Problems
Find the LCM and HCF for the following number pairs using prime factorization:
1. 20 and 30
2. 14 and 21
3. 27 and 36
4. 48 and 72
5. 50 and 75
6. 32 and 40
7. 54 and 90
8. 63 and 84
9. 96 and 108
10. 100 and 125
Answer Key (Click to Show)
1. HCF = 10, LCM = 60
2. HCF = 7, LCM = 42
3. HCF = 9, LCM = 108
4. HCF = 24, LCM = 144
5. HCF = 25, LCM = 150
6. HCF = 8, LCM = 160
7. HCF = 18, LCM = 270
8. HCF = 21, LCM = 252
9. HCF = 12, LCM = 864
10. HCF = 25, LCM = 500