LCM & HCF

"LCM and HCF are crucial concepts in number theory. These concepts are important for finding common multiples and factors between two or more numbers."

1. Basic Concepts and Formulas

  • LCM: Least Common Multiple of two or more numbers is the smallest number that is a multiple of each of them.
  • HCF: Highest Common Factor of two or more numbers is the greatest number that divides all of them.
  • LCM × HCF = Product of two numbers
  • Formula for LCM: LCM(a, b) = (a × b) / HCF(a, b)
  • Formula for HCF: HCF(a, b) = (a × b) / LCM(a, b)
  • Prime Factorization: Decompose numbers into prime factors to easily find HCF and LCM.

2. Types of Questions

  • Type 1: Basic LCM and HCF Calculation
    Example: Find the LCM and HCF of 24 and 36.
    Solution:
    Prime factorization of 24 = 2³ × 3
    Prime factorization of 36 = 2² × 3²
    HCF = 2² × 3 = 12
    LCM = 2³ × 3² = 72
  • Type 2: Relationship Between LCM and HCF
    Example: If LCM of two numbers is 180 and their HCF is 12, and one of the numbers is 36, find the other number.
    Solution:
    Using the formula: LCM × HCF = Product of the two numbers
    180 × 12 = 36 × x ⇒ x = 60
    Therefore, the other number is 60.
  • Type 3: LCM of Multiple Numbers
    Example: Find the LCM of 8, 12, and 15.
    Solution:
    Prime factorization of 8 = 2³
    Prime factorization of 12 = 2² × 3
    Prime factorization of 15 = 3 × 5
    LCM = 2³ × 3 × 5 = 120
  • Type 4: HCF of Multiple Numbers
    Example: Find the HCF of 16, 24, and 48.
    Solution:
    Prime factorization of 16 = 2⁴
    Prime factorization of 24 = 2³ × 3
    Prime factorization of 48 = 2⁴ × 3
    HCF = 2³ = 8
  • Type 5: HCF Using Euclid's Algorithm
    Example: Find the HCF of 56 and 98 using Euclid’s algorithm.
    Solution:
    Step 1: 98 ÷ 56 = 1 (remainder 42)
    Step 2: 56 ÷ 42 = 1 (remainder 14)
    Step 3: 42 ÷ 14 = 3 (remainder 0)
    Therefore, HCF = 14
  • Type 6: LCM of Numbers with Different Prime Factors
    Example: Find the LCM of 5, 9, and 11.
    Solution:
    LCM = 5 × 9 × 11 = 495
  • Type 7: LCM and HCF with Fractions
    Example: Find the LCM and HCF of 3/4 and 5/6.
    Solution:
    LCM of fractions = LCM of numerators / HCF of denominators.
    LCM = LCM(3, 5) / HCF(4, 6) = 15 / 2 = 7.5
    HCF = HCF(3, 5) / LCM(4, 6) = 1 / 12 = 1/12
  • Type 8: LCM and HCF of Decimal Numbers
    Example: Find the LCM and HCF of 0.6 and 1.2.
    Solution:
    Convert to integers: 0.6 = 6/10, 1.2 = 12/10.
    Now, LCM and HCF of 6 and 12: HCF = 6, LCM = 12.
    Therefore, LCM = 12/10 = 1.2, HCF = 6/10 = 0.6
  • Type 9: Sum of LCM and HCF
    Example: Find the sum of LCM and HCF of 18 and 24.
    Solution:
    Prime factorization of 18 = 2 × 3²
    Prime factorization of 24 = 2³ × 3
    HCF = 2 × 3 = 6
    LCM = 2³ × 3² = 72
    Sum = 72 + 6 = 78
  • Type 10: LCM and HCF from Factorization
    Example: Find the LCM and HCF of 48 and 180 by prime factorization.
    Solution:
    Prime factorization of 48 = 2⁴ × 3
    Prime factorization of 180 = 2² × 3² × 5
    HCF = 2² × 3 = 12
    LCM = 2⁴ × 3² × 5 = 720

3. Previous Year Exam Questions

  • (SSC CGL 2021): Find the LCM and HCF of 54 and 72.
    Solution: HCF = 18, LCM = 216
  • (RRB NTPC 2020): If LCM of two numbers is 360 and their HCF is 12, and one number is 60, find the other number.
    Solution: Other number = (360 × 12) / 60 = 72
  • (SSC CHSL 2019): Find the LCM of 15, 25, and 30.
    Solution: LCM = 150

    • 3. More Types of Questions

    • Type 11: LCM of Large Numbers
      Example: Find the LCM of 54 and 90.
      Solution:
      Prime factorization of 54 = 2 × 3³
      Prime factorization of 90 = 2 × 3² × 5
      LCM = 2 × 3³ × 5 = 450
    • Type 12: HCF of Large Numbers
      Example: Find the HCF of 56 and 84.
      Solution:
      Prime factorization of 56 = 2³ × 7
      Prime factorization of 84 = 2² × 3 × 7
      HCF = 2² × 7 = 28
    • Type 13: LCM and HCF with Powers
      Example: Find the LCM and HCF of 12 and 30.
      Solution:
      Prime factorization of 12 = 2² × 3
      Prime factorization of 30 = 2 × 3 × 5
      HCF = 2 × 3 = 6
      LCM = 2² × 3 × 5 = 60
    • Type 14: LCM of Three Numbers
      Example: Find the LCM of 15, 20, and 30.
      Solution:
      LCM of 15, 20, and 30 = 60
    • Type 15: HCF of Three Numbers
      Example: Find the HCF of 18, 24, and 48.
      Solution:
      HCF = 6
    • Type 16: LCM Using Prime Factorization
      Example: Find the LCM of 80 and 120 using prime factorization.
      Solution:
      Prime factorization of 80 = 2⁴ × 5
      Prime factorization of 120 = 2³ × 3 × 5
      LCM = 2⁴ × 3 × 5 = 240
    • Type 17: HCF Using Prime Factorization
      Example: Find the HCF of 36 and 72 using prime factorization.
      Solution:
      Prime factorization of 36 = 2² × 3²
      Prime factorization of 72 = 2³ × 3²
      HCF = 2² × 3² = 36
    • Type 18: LCM and HCF of Different Powers
      Example: Find the LCM and HCF of 12³ and 30².
      Solution:
      Prime factorization of 12³ = (2² × 3)³ = 2⁶ × 3³
      Prime factorization of 30² = (2 × 3 × 5)² = 2² × 3² × 5²
      HCF = 2² × 3² = 36
      LCM = 2⁶ × 3³ × 5² = 72,000
    • Type 19: LCM and HCF for Large Numbers
      Example: Find the LCM and HCF of 56 and 210.
      Solution:
      Prime factorization of 56 = 2³ × 7
      Prime factorization of 210 = 2 × 3 × 5 × 7
      HCF = 2 × 7 = 14
      LCM = 2³ × 3 × 5 × 7 = 840
    • Type 20: Application of LCM and HCF in Time and Work Problems
      Example: A person can complete a work in 12 days and another person in 15 days. What is the least number of days in which both can complete the work together?
      Solution:
      LCM of 12 and 15 = 60. Therefore, both can complete the work together in 60 days.
    • Type 21: LCM of Two Numbers with Multiple Common Factors
      Example: Find the LCM of 12 and 20.
      Solution:
      Prime factorization of 12 = 2² × 3
      Prime factorization of 20 = 2² × 5
      LCM = 2² × 3 × 5 = 60
    • Type 22: HCF of Two Numbers with Common Prime Factors
      Example: Find the HCF of 48 and 60.
      Solution:
      Prime factorization of 48 = 2⁴ × 3
      Prime factorization of 60 = 2² × 3 × 5
      HCF = 2² × 3 = 12
    • Type 23: LCM of Two Numbers with No Common Factors
      Example: Find the LCM of 7 and 9.
      Solution:
      LCM = 7 × 9 = 63
    • Type 24: HCF of Two Numbers with Different Prime Factors
      Example: Find the HCF of 14 and 35.
      Solution:
      Prime factorization of 14 = 2 × 7
      Prime factorization of 35 = 5 × 7
      HCF = 7
    • Type 25: LCM and HCF for Multiple Fractions
      Example: Find the LCM and HCF of 5/9 and 7/12.
      Solution:
      LCM = (LCM of numerators) / (HCF of denominators)
      LCM(5, 7) = 35, HCF(9, 12) = 3
      LCM of fractions = 35/3, HCF = 1/3
    • Type 26: HCF and LCM for Prime Numbers
      Example: Find the LCM and HCF of 7 and 13.
      Solution:
      As both are prime numbers, HCF = 1, LCM = 7 × 13 = 91
    • Type 27: LCM and HCF for Decimal Numbers
      Example: Find the LCM and HCF of 0.5 and 1.5.
      Solution:
      Convert to integers: 0.5 = 1/2, 1.5 = 3/2.
      LCM = 3, HCF = 1
    • Type 28: Using LCM and HCF to Find the Numbers
      Example: The LCM of two numbers is 36 and their HCF is 6. One number is 12. Find the other number.
      Solution:
      Using LCM × HCF = Product of the two numbers: 36 × 6 = 12 × x ⇒ x = 18
      Therefore, the other number is 18.
    • Type 29: LCM and HCF in Ratio Problems
      Example: The ratio of two numbers is 3:4 and their HCF is 6. Find the LCM.
      Solution:
      LCM = (Product of the numbers) / HCF = (3 × 4 × 6) / 6 = 12
    • Type 30: LCM and HCF in Application Problems
      Example: Two bells ring at intervals of 30 seconds and 40 seconds. If they ring together at 12:00 PM, when will they ring together next?
      Solution:
      LCM of 30 and 40 = 120 seconds = 2 minutes.
      Therefore, they will ring together again at 12:02 PM.
StartPreparation