Free GCD LCM Calculator - Find Greatest Common Divisor & Least Common Multiple

Calculate the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of multiple numbers instantly with our free online calculator. Get step-by-step solutions using the Euclidean algorithm and understand the mathematical process behind finding GCD and LCM.

Whether you're a student learning number theory, a teacher preparing lessons, or anyone working with fractions and mathematical operations, our GCD LCM calculator provides accurate results with detailed explanations to help you understand the concepts.

Calculate GCD and LCM

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Understanding the Euclidean Algorithm

How the Algorithm Works

Step 1

Divide the larger number by the smaller number and find the remainder.

Step 2

Replace the larger number with the smaller number and the smaller number with the remainder.

Step 3

Repeat until the remainder is zero. The last non-zero remainder is the GCD.

Example: GCD(48, 18)

48 = 18 × 2 + 12

18 = 12 × 1 + 6

12 = 6 × 2 + 0

GCD = 6

LCM Calculation

LCM(48, 18) = |48 × 18| / GCD(48, 18)

LCM(48, 18) = 864 / 6

LCM = 144

Important Properties and Tips

GCD Properties

  • • GCD(a, 0) = a
  • • GCD(a, b) = GCD(b, a)
  • • GCD(a, 1) = 1
  • • GCD(a, a) = a

LCM Properties

  • • LCM(a, 1) = a
  • • LCM(a, b) = LCM(b, a)
  • • LCM(a, a) = a
  • • LCM ≥ max(a, b)

Quick Tips

  • • If numbers are coprime, GCD = 1
  • • For coprime numbers, LCM = a × b
  • • Always: GCD ≤ min(a, b)
  • • Use for fraction simplification

Real-World Applications of GCD and LCM

GCD Applications

Simplifying Fractions

To reduce 24/36 to lowest terms, find GCD(24, 36) = 12. Then divide: 24÷12 / 36÷12 = 2/3

Tiling Problems

Find the largest square tile that can perfectly tile a rectangular area by finding the GCD of length and width.

Distribution Problems

Distribute items equally among groups by finding the GCD of quantities to determine maximum group size.

LCM Applications

Adding Fractions

To add 1/4 + 1/6, find LCM(4, 6) = 12 for common denominator: 3/12 + 2/12 = 5/12

Scheduling Problems

If events repeat every 6 and 8 days respectively, they'll coincide every LCM(6, 8) = 24 days.

Gear Systems

Two gears with different teeth numbers will align after LCM rotations, useful in mechanical engineering.