Quick Answer: How to Find LCM and HCF
The Highest Common Factor (HCF) is the greatest divisor shared by all given numbers, while the Lowest Common Multiple (LCM) is the smallest multiple shared by all given numbers. For two numbers 24 and 36: • Prime factors of 24 = 2³ × 3¹ • Prime factors of 36 = 2² × 3² • HCF = product of lowest powers of common prime factors: 2² × 3¹ = 4 × 3 = 12. • LCM = product of highest powers of all prime factors: 2³ × 3² = 8 × 9 = 72. Notice the universal law for two numbers: HCF(24, 36) × LCM(24, 36) = 12 × 72 = 864, which exactly matches 24 × 36 = 864. Calculate results for 2 to 6 numbers with our free [LCM and HCF Calculator](/lcm-hcf-calculator).
Method 1: Prime Factorization Method
Prime factorization is the most intuitive method for understanding why LCM and HCF work: 1. Express each number as a product of prime powers (e.g. 60 = 2² × 3¹ × 5¹; 84 = 2² × 3¹ × 7¹). 2. For HCF: Take only the common prime factors, each raised to its lowest exponent present: 2² × 3¹ = 12. 3. For LCM: Take every prime factor that appears in any number, raised to its highest exponent: 2² × 3¹ × 5¹ × 7¹ = 4 × 3 × 5 × 7 = 420.
Method 2: Euclidean Algorithm for HCF (Fastest for Large Numbers)
For large numbers where prime factorization is tedious, the Euclidean algorithm finds HCF in logarithmic steps using repeated division: Example: Find HCF of 252 and 105: • Step 1: 252 ÷ 105 = 2 remainder 42 • Step 2: 105 ÷ 42 = 2 remainder 21 • Step 3: 42 ÷ 21 = 2 remainder 0 The last non-zero remainder is 21. Therefore, HCF(252, 105) = 21! Once HCF is known, LCM is easily found: LCM = (252 × 105) / 21 = 26,460 / 21 = 1,260.
Worked Example with 3 Numbers: 18, 24, and 30
Let us find the HCF and LCM of three numbers: • 18 = 2¹ × 3² • 24 = 2³ × 3¹ • 30 = 2¹ × 3¹ × 5¹ 1. HCF: Common primes across all three are 2 and 3. Lowest powers: 2¹ and 3¹. HCF(18, 24, 30) = 2 × 3 = 6. 2. LCM: All prime factors present: 2, 3, 5. Highest powers: 2³, 3², 5¹. LCM(18, 24, 30) = 8 × 9 × 5 = 360.
Common Traps and Mathematical Rules
• The formula HCF(a, b) × LCM(a, b) = a × b holds strictly for TWO numbers. It does not apply directly to three or more numbers! • Co-prime numbers are numbers whose HCF is 1 (e.g. 8 and 15). The LCM of two co-prime numbers is always their direct product (8 × 15 = 120). • HCF of fractions = HCF of numerators / LCM of denominators. • LCM of fractions = LCM of numerators / HCF of denominators.