GCD and LCM Calculator with Step-by-Step Solutions
Calculate Greatest Common Divisor and Least Common Multiple instantly with detailed explanations
Our free online tool helps students, teachers, and professionals find GCD and LCM values quickly while understanding the underlying mathematical concepts through step-by-step solutions.
GCD and LCM Calculator
2. Replace 98 with 56 and 56 with 42 → GCD(56, 42)
3. Divide 56 by 42 → 56 ÷ 42 = 1 remainder 14
4. Replace 56 with 42 and 42 with 14 → GCD(42, 14)
5. Divide 42 by 14 → 42 ÷ 14 = 3 remainder 0
6. GCD is 14 when remainder becomes 0
LCM(a, b) = (a × b) / GCD(a, b)
2. Multiply numbers: 12 × 18 = 216
3. Divide by GCD: 216 ÷ 6 = 36
4. LCM is 36
About GCD and LCM
Also known as the Greatest Common Factor (GCF) or Highest Common Factor (HCF).
LCM is particularly useful for finding common denominators in fractions.
- Simplifying fractions to their lowest terms
- Distributing items equally into groups
- Cryptography algorithms
- Engineering calculations for gear ratios
- Finding common denominators when adding fractions
- Scheduling repeating events
- Determining when astronomical events will coincide
- Manufacturing production cycles
The concept of LCM has been used since ancient times for astronomical calculations and calendar systems.
Why Use Our GCD and LCM Calculator?
Detailed Steps
See exactly how calculations are performed with step-by-step explanations.
Fast & Accurate
Get instant results using optimized algorithms for GCD and LCM calculations.
Educational
Perfect for students learning number theory and algebra concepts.
Handles Large Numbers
Calculates GCD and LCM for very large integers efficiently.
Common GCD & LCM Applications
Fraction Simplification
Use GCD to reduce fractions to their simplest form:
Event Scheduling
Use LCM to find when repeating events will coincide:
Engineering Ratios
GCD helps determine optimal gear ratios and mechanical configurations:
Practice GCD/LCM Problems
Problem 1: Find GCD(36, 84) and LCM(36, 84)
Advanced GCD/LCM Techniques
Extended Euclidean Algorithm
Finds integers x and y such that ax + by = gcd(a,b). Essential for modular inverses in cryptography.
Multiple Number GCD/LCM
Calculate GCD/LCM for more than two numbers by iteratively applying the operations.
Binary GCD Algorithm
More efficient for computers using bitwise operations instead of division.
GCD & LCM Quick Reference
GCD Basics
- Definition: Largest number dividing both inputs
- Notation: GCD(a,b) or GCF(a,b)
- Example: GCD(12,18) = 6
LCM Basics
- Definition: Smallest number both inputs divide into
- Formula: LCM(a,b) = (a×b)/GCD(a,b)
- Example: LCM(4,6) = 12
Common GCD/LCM Mistakes to Avoid
Confusing GCD with LCM
GCD finds the greatest common divisor, while LCM finds the least common multiple. They solve different problems.
Stopping Too Early
When using the Euclidean algorithm, continue until the remainder is exactly 0, not just close to zero.
GCD/LCM FAQs
GCD (Greatest Common Divisor) and HCF (Highest Common Factor) are different names for the same mathematical concept – the largest number that divides two numbers without remainder.
No, GCD cannot exceed the smaller of the two numbers. By definition, it must divide both numbers.
Our calculator automatically uses absolute values since GCD/LCM are defined for positive integers. The sign doesn’t affect the result.
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