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GCD and LCM Calculator: Greatest Common Divisor & Least Common Multiple

Learn how to calculate GCD and LCM, understand their applications, and master these fundamental mathematical concepts.

2024-12-284 min

What is GCD?

The Greatest Common Divisor (GCD) is the largest positive integer that divides two or more numbers without a remainder.

Example: GCD of 12 and 18

  • Divisors of 12: 1, 2, 3, 4, 6, 12
  • Divisors of 18: 1, 2, 3, 6, 9, 18
  • Common: 1, 2, 3, 6
  • GCD = 6

    • What is LCM?

    The Least Common Multiple (LCM) is the smallest positive integer that is divisible by two or more numbers.

    Example: LCM of 4 and 6

  • Multiples of 4: 4, 8, 12, 16, 20, 24...
  • Multiples of 6: 6, 12, 18, 24...
  • Common: 12, 24...
  • LCM = 12

    • Calculation Methods

    Euclidean Algorithm for GCD

    GCD(a, b) = GCD(b, a mod b)
Stop when b = 0, GCD = a

    GCD-LCM Relationship

    LCM(a, b) = (a × b) / GCD(a, b)

    Real-World Applications

    GCD Uses

  • Simplifying fractions
  • Dividing items into equal groups
  • Finding common patterns
  • Cryptography (RSA algorithm)

    • LCM Uses

  • Finding common denominators
  • Scheduling recurring events
  • Gear ratios
  • Music timing

    • Example Problems

    Simplify 24/36

    GCD(24, 36) = 12

    24/36 = (24÷12)/(36÷12) = 2/3

    When will buses arrive together?

    Bus A: every 15 min, Bus B: every 20 min

    LCM(15, 20) = 60 minutes

    Our GCD/LCM Calculator

  • Calculate for multiple numbers
  • Show step-by-step solution
  • Prime factorization method
  • Euclidean algorithm display
  • Instant results